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Groupings and Distributions
by Total Gadha - Wednesday, 29 August 2007, 06:27 AM
 

cat 2007 cat 2008 mba 2008 xat 2008 permutation and combinationI have been itching to write this particular article containing higher order fundamentals of permutation and combination. Although CAT 2007/ CAT 2008 aspirants may take a little time to digest this article, the article will clear a lot of doubts in this area. This particular area of grouping and distribution has been my bugaboo for many years until recently when I sat down and unraveled one thread of the mystery after another; the joys of discovering at leisure. Please do not fail to ask me any question that occurs to you after reading this article.

Number of ways of grouping dissimilar things

How would you divide seven different objects in two groups of four and three? Simple. You select four things out of those seven and three will be left behind. Or you can select three things out of those seven and four will be left behind. The number of ways you can accomplish this is 7C4 or 7C3. Now, how would you divide these seven different objects in three groups of one, two and four? Again, you can begin by choosing four things out of these seven, then two things out of the remaining three. The number of ways you can do this is 7C4 Ã— 3C2. Therefore, to divide things into groups, you keep on selecting groups, except for the last group which will be automatically formed.

So how would you divide seven different things into groups of three, three and one?

Your answer would be to take our three things first, then three things next, i.e. 7C3 × 4C3 = 140, and your answer would be WRONG!

Why is your answer wrong? Is there anything wrong with the method? No. Only that you need to take some extra precautions when you are making equal groups. There are slight changes to be observed when you divide some number of things into equal groups.

Let’s start small. According to the method above, in how many ways can you divide 4 different things (say a, b, c and d) into two groups having two things each? Your answer would be to select two things out of the four and two would be left behind, i.e. 4C2 = 6. But are there really 6 ways?

Given below are shown the number of ways we can divide four things, a, b, c and d, into two groups of two:

CAT 2007 CAT 2008 mba 2008 XAT 2008 permutation and combination

You can keep trying if you want to, but there are only 3 ways of dividing the four things into two groups of two.

Where did the rest of 3 ways calculated through 4C2 = 6 disappear?

The answer is that they got merged. When you selected two things out of the four, the things selected were ab, ac, ad, bc, bd, and cd. But the last three groups are already formed when you select the first three groups, i.e. when you select ab, you automatically get cd. When you select ac, you automatically get bd, and so on.

Let’s do it again. According to the method above, in how many ways can you divide 6 different things (say a, b, c, d, e and f) into three groups having two things each? Your answer would be to select two things out of those six and then select two things from the remaining four so that two would be left behind, i.e. 6C2  × 4C2 = 90. But are there really 90 ways?

The table below shows the number of ways you can divide six different things into three groups of two each:

CAT 2007 CAT 2008 MBA 2008 XAT 2008 permutation and combination


Example: How would you divide 5 distinct objects into groups of 2, 2 and 1?

Answer: The single object can be chosen in 5C1 = 5 ways. The rest of the 4 objects can be divided into two equal groups in 3 ways. Therefore, the number of ways = 5 Ã— 3 = 15.

Note that the answer by our first method will be 5C1 × 4C2 = 30 ways. This answer is incorrect.

Number of ways of grouping similar things

In how many ways can you divide five similar objects (say a, a, a, a, a) into three different groups? When we make groups of similar objects, the only way we can differentiate two different ways of grouping is by differentiating between groups when they have different size (number of objects). Therefore, in case of similar objects, the number of different ways we can group them is the number of different sizes of groups that we can make. Let’s see how many groupings of different sizes we can make for 5 similar objects.

 cat permutation and combination

DISTRIBUTION

After you have made groups of some objects, you might want to distribute these groups in various places. For example, after you made groups of some toffees, you might want to distribute these groups among some children. Or, after dividing some number of balls into groups, you might want to distribute these groups into boxes. Just as the objects that we group can be similar or dissimilar, so can the places that we assign these groups to be similar or dissimilar. While distributing groups, we need to keep one rule in mind:

cat

Now let’s solve an all-encompassing example on what we have learnt.

Question: In how many ways can you put 5 balls in 3 boxes if

I.                    the boxes are similar and the balls are similar.
II.                 the boxes are different but the balls are similar.
III.               the boxes are similar but the balls are different.
IV.               
the boxes are different and the balls are different.

Answer: Sums up everything, doesn’t it? Well, here we go:

Case I: the boxes are similar and the balls are similar

cat 

Case II: the boxes are different but the balls are similar

cat

cat

Case III: the boxes are similar but the balls are different

cat

Case IV: the boxes are different and the balls are different

cat
Re: Groupings and Distributions
by Anil Sahu - Wednesday, 29 August 2007, 08:18 AM
 

Hi TG,

I was longing to see such kind of post on P & C. And I think this is the perfect post about P & C which gives a gist of all the stuffs. I am sure it will be of great help to all MBA aspirants. Keep posting dude. smile

I would love to see similar posts on other topics like Inequalities, quadratic equations, graphs etc as well.

thank you so much in advance.

-Anil

Re: Groupings and Distributions
by lavika gupta - Wednesday, 29 August 2007, 08:21 AM
  hi tg.this is a wonderful article.this was a weaker area of mine,but your article has helped clarify the concept.three cheers 2 u.
Re: Groupings and Distributions
by the underdog - Wednesday, 29 August 2007, 08:26 AM
  Excellent!!! I wish we could stop time so that we'd have ALL your wonderful articles before CAT 07! smile
Re: Groupings and Distributions
by rajesh ranjan - Wednesday, 29 August 2007, 09:59 AM
  your relentless marvellous work will help many cat aspirants in their preparations......................thanx dude.keep up the good work !!
Re: Groupings and Distributions
by chandrasekhar prasad munu - Wednesday, 29 August 2007, 11:09 AM
 

Hi TG

Please explain how 1 1 and 3 groupings  can be distributed in 3! ways as 2 groupings are similar.This should be 3!/2! ways.same for 1 2 2.

Re: Groupings and Distributions
by dinesh munna - Wednesday, 29 August 2007, 02:12 PM
  Sirrrrrrrrr... I must admit that i have never seen an article as simply and beautifully written and as informative as the one u just posted...to put it in one word - its a COMPREHENSIVE account of all that one needs to know about groupings...thanx a tonnn....A small request to u sir..if u dont mind....Can u add an article on SETS - minimisation and maximisation stuff type questions which occur frequently in DI sets pls???I became such an ardent fan of TG.COM that i had to set it as my homepage!!!believe me , u r just awesummmm...smile
Re: Groupings and Distributions
by ankur gur - Wednesday, 29 August 2007, 02:49 PM
 

Hi TG,

As in the example we have been enlisting all the possible groupings for a set of 5 balls into three sets can we have formula to find the number of groupings?? B'coz incase the number of balls are large lets say dividing 40 balls into 3 groups than what will be the number of possible sets???

Re: Groupings and Distributions
by Kunal Gupta - Wednesday, 29 August 2007, 03:16 PM
 

Hi TG,

Tk pains to explain following

case : the boxes are different but the balls are similar

in 0 1 4 case how the distribution is 3!??

I cdnt understand hence coudnt proceed..

Re: Groupings and Distributions
by saurabh yadav - Wednesday, 29 August 2007, 03:50 PM
  HI TG,
marvellous, bombastic,savoury,didactic,awesome all these adjectives go for this article. TG thanks for taking pain and i wish more article come on the way before cat.

THANKS,
SAURABH YADAV
Re: Groupings and Distributions
by Pawan Raghuveer - Wednesday, 29 August 2007, 06:45 PM
 

Hi TG,

Brilliant boss! This is my first visit to the site and already i think its gonna be very useful! Hats off to the your illustrations.

Cheers

 

Re: Groupings and Distributions
by Sri KLR - Wednesday, 29 August 2007, 09:52 PM
 

Another feather in TG's cap. Another lesson in our lap !!

I still can't believe that no.of ways distributing 5 similar things in 3 similar boxes is jus 5. Gosh!! what would have happened if it came in Cat this year and I hadn't gone thru this lesson....I would have been frantically searching for 5C2 in the options !!

Re: Groupings and Distributions
by vishal grover - Wednesday, 29 August 2007, 10:03 PM
 

fabulous artice....the timing is just perfect...i was looking around for someone to clear the distribution similar/dissimilar fundas....

must admit dats a v neat article ...kudos..!!

Re: Groupings and Distributions
by Anupam Agarwal - Thursday, 30 August 2007, 12:41 AM
 

Ur intelligence and ur creativity is makin me jealous...the way u xplain and the way u make difficult thins so simpler..thas really awesome..

Re: Groupings and Distributions
by amit ahuja - Thursday, 30 August 2007, 12:44 AM
 

hi tg,

thanks for the explanation in such a clear manner....

why dont u start with basics of p and c, i.e all fundas dealing with p and c like dearrangements and other things like circular permutations.....

also kindly write an article on probability (this is the last chapter according to preference given by cat takers, which is yet unwinded from ur side).....

I request u to do the same......

thanks again for such a nice and condensed article....

regards

Amit Ahuja

 

 

 

Re: Groupings and Distributions
by divya kansal - Thursday, 30 August 2007, 09:45 AM
 

Gr8 article...

the best thing abt TG is that he explains complex things in a simple manner

which even the coaching instis are unable to do. Hats off TG!!

Re: Groupings and Distributions
by Sachin Joshi - Thursday, 30 August 2007, 10:12 AM
  Really good article. You are doing a great job and please try and get in as many such articles as possible before CAT 07.
Re: Groupings and Distributions
by vishal grover - Thursday, 30 August 2007, 12:44 PM
  Hey TG i ve another doubt..maybe an extension of the 4 cases....if in the 4 th case ( distributing Different => Different) say the internal arrangement is also important....eg distributing 4 different balls to 3 different funnels wherein the balls within a funnel could themselves be internally arranged....wat wud be approach n shortcut for the same...??
Re: Groupings and Distributions
by don don - Thursday, 30 August 2007, 07:11 PM
 

Just awesom!

 

Re: Groupings and Distributions
by rmmozhi prathiba - Friday, 31 August 2007, 06:32 AM
 

hi tg it was wonderful.. i have a doubt how to solve the following problem:

if the numbers 2^218 and 3^109 are both evaluated converted into base 12 and written one beside the other, how many digits are there, on the whole, in the result??

 

Re: Groupings and Distributions
by arun kumar - Friday, 31 August 2007, 03:39 PM
  sir ,
    in the case where the boxes r different and the balls r different, u have considered the no of ways for the distribution of 1,1,3 and 1,2,2 as 3!=6 but shouldnt it be 3 ....am i missing something ????
thoughtful
Re: Groupings and Distributions
by Santosh Ram - Friday, 31 August 2007, 11:14 PM
 

hi Mr. TG,

Thanks a million for posting such an excellent article for MBA aspirants.

I wud say that u r Totally Gr8 !!!smile

Posting such articles in other Topics as well will well be received amongst all.

Cool,

Santosh

Re: Groupings and Distributions
by Anant Garg - Sunday, 2 September 2007, 10:26 PM
 

HI TG,

pl's find me the way to solve these P & C probs:

1. The first n natural numbers, 1 to n ,have to be arranged in a row from left to right. In how many ways can it be done, if n=7 and there are an odd number of numbers between any two even numbers and an even number of number between 1 and 2 ?

2. In how many ways it can be done, if  n = 7 and there are even number of numbers between any two odd numbers and an odd number of numbers between 1 and 2? 

TIA,

Anant

Re: Groupings and Distributions
by Fics Ics - Monday, 3 September 2007, 08:32 AM
  " sir ,
    in the case where the boxes r different and the balls r different, u have considered the no of ways for the distribution of 1,1,3 and 1,2,2 as 3!=6 but shouldnt it be 3 ....am i missing something ????
thoughtful   "

I agree with Arun...
are we missing something???? mixed
Re: Groupings and Distributions
by Total Gadha - Monday, 3 September 2007, 10:27 AM
  Hi Fics,

is the case (a, b, cde) same as (b, a, cde)?

Total Gadha
Re: Groupings and Distributions
by Fics Ics - Monday, 3 September 2007, 08:08 PM
  Thankyou sir smile

Point noted wide eyes
Re: Groupings and Distributions
by chaitanya santosh - Thursday, 6 September 2007, 10:51 AM
  sir can u please clarify on this point
Re: Groupings and Distributions
by the underdog - Saturday, 8 September 2007, 09:48 PM
 

Tg Sir!

Could you please explain the following:

Dividing 500 students into five distinctly identifiable sections of 100 students each = 500!/(100!)5. Here why don't we have 5! in the denominator? Is it because of "distinctly identifiable sections"? Does your formula apply to groups which are not distinctly identifiable?

Re: Groupings and Distributions
by Total Gadha - Sunday, 9 September 2007, 07:18 AM
  Hi Underdog,

Distinctly identifiable 'sections' means that groups are also to be arranged in 5! ways. For example, you can divide 500 toffees in 5 groups of 100 each in 500!/5!(100!)5 ways. Then you can distribute them among 5 boys, giving one group to one boy, 5! ways. Therefore, total number of ways = (500!/5!(100!)5) × 5! = 500!/(100!)5

Total Gadha
Re: Groupings and Distributions
by the underdog - Sunday, 9 September 2007, 09:44 PM
  Thanks a lot TG! Got it. smile. I have one humble request. Could you please provide us a lesson on at most/at least (venn diagram) related problems? I'm not at all confident in it and usually take too many iterations.
Re: Groupings and Distributions
by Satyam Gadha - Monday, 10 September 2007, 08:32 AM
  Hi TG,
 This article is awesome and the last funda of taking partitions as objects and placing them between the objects to get the number of ways objects can be partitioned is now clear after coming to this page. I churned Google before for getting the same funda clear when I was solving a question on other site, but got nothing.

Thanks
SatyaGadha
Re: Groupings and Distributions
by joy das - Thursday, 13 September 2007, 12:50 PM
  Hi TG
The lesson was great.But I hav one quiery....can we find the total number of positive integer  solutions for an equation like
5x+y+z=500..thoughtful  by the above mentioned method.???wide eyes Plz reply asap!
Re: Groupings and Distributions
by garima jain - Friday, 21 September 2007, 06:22 PM
 

DearTG,

plz plz plz post something on probability as well...

plz...

i beg..

except u no one can help me with it.

i would havbe never understood relative speed or pnc for that sake...

teachers r too mechanical..too boring..to formula based...

plz help..

Re: Groupings and Distributions
by sMILING GadhUUU - Saturday, 22 September 2007, 02:11 AM
 

Fabulous Article TG,

Thanking you...............

I bow to TG          

Re: Groupings and Distributions
by Akshun Gulati - Friday, 2 November 2007, 07:54 PM
  Hello TG,

This is the perfect concordance of everything I have studied in "Groupings and Distribution"... All the concepts have been so beautifully interlinked, giving a pan optic view of this topic...

Thanks a ton,

- Akshun
Re: Groupings and Distributions
by Rahul Jena - Thursday, 10 January 2008, 01:30 PM
 

Hi TG,

Me not being able to understand the following question:-

Question: In how many ways can you put 5 balls in 3 boxes if


II.                 the boxes are different but the balls are similar.

Here  doubt is 7C2 , hw it equal to 21.It shd be 42 .Plz correct me if I am missing some condition.

III.               the boxes are similar but the balls are different.
IV.                the boxes are different and the balls are different.

The above two I am not being able to understand anything..how the groupings and distribution is being done...Surry for this query......I know you have written it in the simplest possible manner, but still i am not seeing the concept behind it.Plz explain it as you  wd explain a Gadha student of urs....Explain for any one questioon, then as homework, i will explain the other.


 

Re: Groupings and Distributions
by sathia p - Saturday, 26 July 2008, 03:44 AM
  Hi Tg,
Since it the example that your have explained the number me ball and the groups are less i.e 5 balls and 3 box we where able to list all the possible grouping.but it will become really tough if the question is 60 balls and 10 boxes.is there any general method to find this.

For the 2nd case when the boxes are different and the caller are simillar we have a formula
(n balls & r boxes)
(n+r-1)C(r-1)

For the 4th care when the cords are different and the balls are also different we have a formula
(n balls & r boxes)
r^n
Simillary is there any short cuts for the 1 and the 3 case.

Regards
sathia
Re: Groupings and Distributions
by rajdeep choudhuri - Thursday, 31 July 2008, 11:57 PM
  Please TG Sir include a chapter on Permutation&Combination 
Re: Groupings and Distributions
by SHRESHTH ANAND - Saturday, 2 August 2008, 09:08 AM
  Hi All,
I have a serious doubt in the section mentioned below :

Number of ways of grouping similar things

In how many ways can you divide five similar objects (say a, a, a, a, a) into three different groups? When we make groups of similar objects, the only way we can differentiate two different ways of grouping is by differentiating between groups when they have different size (number of objects). Therefore, in case of similar objects, the number of different ways we can group them is the number of different sizes of groups that we can make. Let’s see how many groupings of different sizes we can make for 5 similar objects.

 cat permutation and combination

In my opinion,The  number of ways of grouping 5 similar things to three different groups should be  21 as explained below:
If there are three different groups A, B and C say then the following cases can be made:
                  
Groupings
Number of ways
 0, 0, 5
3!/2! ways = 3
0,1,4
3! ways = 6
0,2,3
3! ways = 6
1,1,3
3!/2! ways = 3
1,2,2
3!/2! ways = 3
Total
21 ways

Because the groups are different
Had the groups been similar for eg: Number of ways in which 5 similar balls can be put in 3 similar boxes then it would have been 5.

what say???
please explain????

Cheers
Shreshth

                     
Re: Groupings and Distributions
by Total Gadha - Sunday, 3 August 2008, 08:27 PM
  Hi Shreshtha,

At least what I have written. sad

I have covered this case in "boxes different balls same".


Total Gadha
Re: Groupings and Distributions
by sathia p - Tuesday, 5 August 2008, 08:39 AM
 

HI TG

In the example that you have explained 5 balls and 3 boxes finding the groupings of the ball were easy since the number of balls are less,where as when the number of ball are more say for example 40 balls and 10 boxes its becomes tedious to find the groupings of the ball.Your explanation have cleared the concept but is there any shortcut for solving the 1 and III condition as we have formula for the II and the IV condition.

I           the boxes are similar and the balls are similar.
II.           
      the boxes are different but the balls are similar.  (n+r-1)C(r-1)
III.              
the boxes are similar but the balls are different.
IV.               
the boxes are different and the balls are different. (n balls & r boxes) r^n

Eagerly awating for your replysad


Re: Groupings and Distributions
by Partha Ray - Thursday, 25 September 2008, 11:54 AM
 

Hi TG,

Can you pls throw more light on the approach to this particular type of distribution problem.

There is a guy named Gadhanand...who appeared an exam that consisted of 4 papers...the maximum marks of 3 of these papers is 50...while the max of 4th paper is 100... In how many ways can he secure 60% of marks in the exam in order to change his name from Gadhanand to Vidyanand???

Partha

Re: Groupings and Distributions
by Kosher InBlues - Thursday, 25 September 2008, 11:00 PM
  HI TG ,
          Thanks a lot , you have covered each and every nuances of the particular problem and that is what  keeps me in awe

Re: Groupings and Distributions
by nice Smile - Monday, 1 June 2009, 10:51 PM
  Hi TG,

i am quite a student of yours and i like your teaching methods. i have a problem here that has been puzzling me for quite some time now. here is how the problem goes,

in how many ways can 12 similar chocolates be distributed among 3 persons where each person can get a maximum of 6 chocolates?

a similar problem is as follows,
find the number of four digit numbers that can be generated using the digits 0,1,2.... 9 such that the sum of the digits of the four digit number is 12.
 
I feel the solution to these two should be on the same lines . please let me know, i also welcome other students to gimme the solution if they know.

regards,
a student smile
Re: Groupings and Distributions
by Total Gadha - Tuesday, 2 June 2009, 12:02 AM
  Hi Nice Smile,

In how many ways can 12 similar chocolates be distributed among 3 persons where each person can get a maximum of 6 chocolates?
You can solve this question by using the reverse method for distribution- that is picking. First distribute 6 chocolates each to all the three students. Now they have 18 chocolates in total. Now you need to pick 6 chocolates out of these 3 groups which is the solution of a + b + c = 6 = 8!/6!2! = 28

Find the number of four digit numbers that can be generated using the digits 0,1,2.... 9 such that the sum of the digits of the four digit number is 12.
As the first digit cannot be 0, we require the coefficient of x12 in the product (x + x2 + x3 + ... + x9)(1 + x + x2 + x3 + ... + x9)3 = coefficient of x11 in the product = (1 - x9)/(1 - x) × (1 - x10)3/(1 - x)3 = (1 - x9)(1 - x30 - 3x10 + 3x20)(1 - x)-4 = 332 (expanding (1 - x)-4). I hope the answer is correct.


Total Gadha


Re: Groupings and Distributions
by nice Smile - Tuesday, 2 June 2009, 02:54 PM
  Hi TG,

I liked the solution to the first question. Thank u so much, i never thought about reverse distribution. now i m enlightened with this too smile
i did not understand ur solution to this question:

Find the number of four digit numbers that can be generated using the digits 0,1,2.... 9 such that the sum of the digits of the four digit number is 12.

wy do we take the co-efficient of x^11 as the answer. i do not understand wy are u multiplying those two expressions in the first place sad . please explain the logic behind it.

but i did try another way to find the answer and it seems correct to me. please verify.

the sum of the numbers should be 12 and i consider them as 12 balls. and since it is a 4 digit number i consider it as four different baskets(a,b,c,d). Now, the first digit cannot be zero so i put one ball in the first basket(a){let me call the rest of the balls that go to basket one as a'}. therefore the problem reduces to the number of ways of dividing these 11 balls into the four baskets with three important caveats:
1. all the cases where all the 11 balls have been put in the same basket have to be excluded
2. all the cases where 10 balls have been put in the same basket have to be excluded
3. all the cases where 9 balls have been put in the first basket has to be excluded(since it ll make the value as 10 for the first basket a, this is invalid since the maximum that we can put in a basket is 9).
Now for the calculations,
no. of ways of dividing 11 balls among 4 baskets is:
a'+b+c+d = 11, (n+r-1)C (r-1)= (11+4-1)C (4-1) = 14 C 3 = 364

Now for case 1: if all eleven are put into one basket it is wrong since the maximum that can be there in a basket is 9, therefore we exclude this. this is nothing but arranging 11,0,0,0 which is equal to
4!/3! = 4 ways

similarly for case 2: this is nothing but arranging 10,1,0,0 which is equal to
4!/2! = 6 ways

for case 3: we need to consider only those cases where we have 9 in the first basket, and they are the number of ways of arranging
9,1,1,0 (with nine fixed in the first place) = 3!/2! = 3 ways
9,2,0,0 (with nine fixed in the first place) = 3!/2! = 3 ways

hence the total is 364 - (4+6+3+3) = 364-16 = 348

the answer is different from what you have got, but i feel this should be right. Please confirm.
Re: Groupings and Distributions
by Total Gadha - Tuesday, 2 June 2009, 10:32 PM
  Hi Nice Smile,

Both of us made a mistake, the answer is 342. You have calculates 4!/2! = 6 whereas it is equal to 12.

Total Gadha
Re: Groupings and Distributions
by nice Smile - Tuesday, 2 June 2009, 10:36 PM
  thank you tg, for informing me that my approach is right. i have been waiting the whole day for your reply smile

but i still did not get how you solved it sad ..... and i missed clearing a cut- off once in cat just because of an addition mistake (a silly mistake like this one) ... i have to wait an year more coz of that sad .....

can you please tell me why did you multiply both of those expressions??

Re: Groupings and Distributions
by Total Gadha - Wednesday, 3 June 2009, 09:39 PM
  Hi Nice Smile,

Simply because of the fundamental rule of counting- that if a job can be done in m ways and the other in n ways, both can be done in m × n ways. The expression (x + x2 + x3 + ... + x9) denotes the values 1st digit can take. The expression (1 + x2 + x3 + ... + x9) denotes the values 2nd, 3rd and 4th digit can take.


Total Gadha
Re: Groupings and Distributions
by nice Smile - Friday, 5 June 2009, 08:03 PM
  thank u tg.. i got it now smile
Re: Groupings and Distributions
by TG Team - Saturday, 6 June 2009, 05:26 PM
  Hi TGsmile
Still not getting that expression funda.
Can you elaborate bit more.
Re: Groupings and Distributions
by nice Smile - Sunday, 14 June 2009, 05:52 PM
  hi kamal,

the logic goes like this ....

whenever we are considering the power of x, then it is like chosing one among the digits 0-9, therefore , x^0 stands for 0, x^1 stands for 1 and so on. to get the sum of the digits as 12 we check the co-efficient of x^12 which means that the sums of the terms was 12. This works since the co-efficient of x^12 will have all the combinations of x^n such that it gives the answer. for ex: x^12 can be written as x * (x^9) * (x^2) * (x^0) and this would represent number 1920, where x is from first expression, (x^9) is from second expression and so on. we did not take (x^0) for the first expression since the first digit cannot be zero. hope this clarifies ur doubt.
Re: Groupings and Distributions
by bhupi suhag - Monday, 22 June 2009, 04:06 PM
  In how many ways can 10 identical presents be distributed among 6 children so that each child gets at least one present.....pls solve 
Re: Groupings and Distributions
by shiva subramanian - Monday, 22 June 2009, 04:51 PM
  hello TG sir..i am new to your site and am interested in buying the geometry ebook..what are the topics covered in it sir?is it a comprehensive book for  cat?similar doubts for the numbers e book as well..and regarding the cat cbt club will it include full length mock tests as well?
Re: Groupings and Distributions
by satyarth pandey - Monday, 7 September 2009, 05:00 PM
  hi tg,

can you kindly explain the following case in detail i'm not able to grasp from the given explanation

IV.               
the boxes are different and the balls are different.

thanks
Re: Groupings and Distributions
by satyarth pandey - Tuesday, 8 September 2009, 02:35 AM
  ignore my prev post ....got the logic.....
Re: Groupings and Distributions
by nice Smile - Tuesday, 20 October 2009, 11:17 PM
  Find the number of four digit numbers that can be generated using the digits 0,1,2.... 9 such that the sum of the digits of the four digit number is 12.As the first digit cannot be 0, we require the coefficient of x12 in the product (x + x2 + x3 + ... + x9)(1 + x + x2 + x3 + ... + x9)3 = coefficient of x11 in the product = (1 - x9)/(1 - x) × (1 - x10)3/(1 - x)3 = (1 - x9)(1 - x30 - 3x10 + 3x20)(1 - x)-4 = 342 (expanding (1 - x)-4).

please explain how did you calculate the value 342 .... i really did not understand sad
Re: Groupings and Distributions
by anup dangaich - Wednesday, 1 September 2010, 10:43 PM
  awesomeness redefined..........
kindly post more articles, m not seeing  any new article now plz dont hibernate, we need u in this monsoon
Re: Groupings and Distributions
by Deepak Kumar - Thursday, 14 October 2010, 12:48 PM
 

Hi TG,

In response to your answer to "In how many ways can 12 similar chocolates be distributed among 3 persons where each person can get a maximum of 6 chocolates?"

It's similar to when balls are similar but boxes are different. That way number of distibutions should come a multiple of three( because we are distributing to three people).

But 28 is not multiple of three?

Re: Groupings and Distributions
by Dinesh Bharadwaj - Monday, 8 August 2011, 02:06 AM
  thanks TG sir.really helped.always searched permutation and combination.but now got a clear method of solving these questions on basis of grouping and distribution.
cool thoughtful
Re: Groupings and Distributions
by apurba biswas - Monday, 13 February 2012, 10:09 AM
 

Hi,

It's a nice example on group and distribution. However I have one question on this topic.

In case II : i.e. if the boxes are different but balls are simillar, the formula for finding total no of ways of distribution of n non-distinct objects into r distinct groups is n+r-1 Cr-1.

In case IV :  i.e. if the boxes are different and balls are different, the formula for finding total no of ways of distribution of n distinct objects into r distinct groups is rn.

Is there any straight forward formula for finding out total no of ways of distribution for Case I and Case III (Like Case II and Case IV) ?

 

Re: Groupings and Distributions
by nitish aggarwal - Sunday, 11 March 2012, 07:53 PM
  how can we group 9 members in 3 groups of three thoughtful
Re: Groupings and Distributions
by arsh arora - Monday, 12 March 2012, 04:24 AM
  9!/(3!)^4
Re: Groupings and Distributions
by nitish aggarwal - Monday, 12 March 2012, 06:22 PM
  its not perfectly divisible and the no of ways can not be in fraction!!!!
Re: Groupings and Distributions
by TG Team - Wednesday, 14 March 2012, 12:51 PM
 

Hi Nitish smile

9!/(3!)4 = 280 i.e. an integer. Please check.

Kamal Lohia

Re: Groupings and Distributions
by rachit mongia - Monday, 30 July 2012, 01:02 PM
  Dear Sir,

I am not able to get this concept which you have also asked for

Could you please explain the same?
Re: Groupings and Distributions
by anirudh bhardwaj - Tuesday, 7 August 2012, 04:36 PM
 

Hi Sir,

Please let me know is "Case IV: the boxes are different and the balls are different "

similar to "grouping n different things among r groups =r^n" ?

 

Thanks.

 

Re: Groupings and Distributions
by GAUTHAM thomas - Wednesday, 19 December 2012, 02:46 AM
  In how many ways can 12 similar chocolates be distributed among 3 persons where each person can get a maximum of 6 chocolates?

TG Sir, suppose if this question was changed such that, the first person receives at most 6, second person receives at most 7 and third person receives at most 8. Then this would be about
solving

A+B+C = 12 where A is less than or equal to 6
B is less than or equal to 7
C is less than or equal to 8

Now, what about this case sir? If the technique mentioned by you is used, then we have to distribute 6 to A, 7 to B, 8 to C. So in total they have 6+7+8 = 21 chocolates. So we have to pick 9.
So this is about solving the equation A+B+C = 9. This is by the method of partition gives 11C2 ways or 55 ways.
But the solution mentioned says 45 ways sad

What mistake am I doing ?
Re: Groupings and Distributions
by ankit gaur - Friday, 21 December 2012, 02:25 PM
  Hi Gautam,
I already answered the same problem.
Re: Groupings and Distributions
by GAUTHAM thomas - Tuesday, 25 December 2012, 03:43 PM
  Hi TG Sir and all,

This post clearly explains the following

1. Similar Things -> Different places
2. Similar Things -> Similar places
3. Different Things -> Different places
4. Different Things -> Similar places

There is also a great explanation about
1. Grouping of Different Objects
a. into groups of equal size
b. into groups of UNequal size
2. Grouping of Similar Objects

But,

Can some one pls explain how to GROUP AND DISTRIBUTE cases where
both SIMILAR AND DIFFERENT objects are present. Eg, the case where
Balls -> B1, B1, B1, B2, B3 have to be grouped and distributed ?

Anybody .. pls help me out ! sad
Re: Groupings and Distributions
by GAUTHAM thomas - Tuesday, 25 December 2012, 03:49 PM
  Hi Ankit sir,

Thanks so much for the reply!! ... I've been looking for this explanation all over the place

I have a few doubts from what you said.. for the first question which TG had explained in the forum

1. In the question which TG had solved, A+B+C = 12 where A,B,C can take a maximum value of 6. This means, that A,B,C are all less than or equal to 6 right ?

2. Suppose if I changed the limits to -- > A,B,C are all less than or equal to 5. Even then the technique doesn't give the correct answer. The equations to be solved will be A+B+C=3

I assumed that this technique (mentioned by TG) could be used only if all 3 variables had the same upper limit. Am i wrong in thinking so ? (because in this we are removing a single 3 from each of A,B,C. The upper limits are 5. SO we can definitely remove a 3.. But it doesn't work) sad

3. Can you please explain how you arrive at the variables X,Y,Z ?

4. Can you throw more light on your statement - >
"But in this case method we can not take 9 or 8 or 7 back from x, same for y or z"

Thanks so much for the explanation. Pls help me in clearing this up
Re: Groupings and Distributions
by naidu ar - Tuesday, 30 April 2013, 06:35 PM
  awesome article sir smile

Please suggest me some more articles on Permutations and combinations
Re: Groupings and Distributions
by issac issie - Monday, 29 July 2013, 10:00 PM
  boxes are different...let 0,1,4 be placed in respective boxes...say A,B,C....they can also be put as 0,4,1....1,0,4....1,4,0....4,1,0....4,0,1....0,1,4....like this in A,B,C in 6 ways..i.e. 3!....

Re: Groupings and Distributions
by sravani jaganti - Monday, 23 June 2014, 06:03 PM
 

In how many ways can you divide five similar objects (say a, a, a, a, a) into three different groups? When we make groups of similar objects, the only way we can differentiate two different ways of grouping is by differentiating between groups when they have different size (number of objects). Therefore, in case of similar objects, the number of different ways we can group them is the number of different sizes of groups that we can make. Let’s see how many groupings of different sizes we can make for 5 similar objects.

 cat permutation and combination


sir i have understood till now.But now can u please explain this concept with few more examples .and doesnt this have any formula as like for dissimilar objects .


while coming to distributions i have got the idea that first we need to divide them into different groups and then we can follow your procedure.

But how do we find those groups ...??

0 0 5,0 1 4,.... how did u get this sir ???

hope you reply soon smile

Re: Groupings and Distributions
by TG Team - Tuesday, 24 June 2014, 02:24 PM
  Hi Sravani smile

You are to distribute 5 identical things into 3 groups. Just think of what groupings are possible and write them. See if you get anything other than those mentioned in above post.

To start small, you can just check the ways to distribute 2 identical things (say a, a) in 2 groups. Isn't it, there are 2 ways only i.e. one a in each group i.e. 1, 1 OR both a's in one group and other on empty i.e. 0, 2.

Regarding second query, 'formulae' are also application of some external brain. So why to depend on someone else's brain if you have a one and can apply it. By the way, there is not any direct formula for this so far. smile

Kamal Lohia