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algebra alert
by soumya sawarn - Thursday, 4 September 2014, 12:20 PM
  I have a huge pile of doubts in algebra and so I am posting them here
find the sum of the series 1(1!)+2 (2!)+ .......+n(n!)
Re: algebra alert
by TG Team - Thursday, 4 September 2014, 12:50 PM
  Hi Soumya smile

1(1!) + 2(2!) + 3(3!) + ... + n(n!)
= {2 - 1}(1!) + {3 - 1}(2!) + {4 - 1}(3!) + ... + {(n + 1) - 1}(n!)
= {2(!) - 1(1!)} + {3(2!) - 1(2!)} + {4(3!) - 1(3!)} + ... + {(n + 1)(n!) - 1(n!)}
= {2! - 1!} + {3! - 2!} + {4! - 3!} + ... + {(n + 1)! - n!}
= (n + 1)! - 1 smile

Kamal Lohia
Re: algebra alert
by soumya sawarn - Tuesday, 9 September 2014, 12:23 AM
  S is a set of three numbers whose sum and HCF are 168 and 8 respectively. How many values are possible for S if at least one of the numbers is not more than 24?
Re: algebra alert
by soumya sawarn - Wednesday, 10 September 2014, 09:22 AM
  Please solve this problem
Re: algebra alert
by TG Team - Thursday, 11 September 2014, 12:31 PM
  Hi Soumya smile

Let the three numbers be a, b, c such that a = hx, b = hy and c = hz where HCF (a, , b, c) = h and HCF (x, y, z) = 1

Now it is given that h = 8 and a + b + c = 168.
So 8x + 8y + 8z = 168
i.e. x + y + z = 21

Further it is given that at least one of a, b, c is not more than 24 i.e. one of x, y, z is not more than 3.

Let x be that number, whose value can not be more than 3.

Now just make the cases:
(i) If x = 1, then y + z = 20 i.e. 10 cases {(1, 1, 19), (1, 2, 18), ..., (1, 10, 10)}
(ii) If x = 2, then y + z = 19 i.e. 7 cases {(2, 2, 17), (2, 3, 16), ...., (2, 8, 9)}
(iii) If x = 3, then y + z =  18 i.e. 4 cases {(3, 4, 14), (3, 5, 13), (3, 7, 11), (3, 8, 10)}

Thus exactly 10 + 7 + 4 = 21 sets of three such numbers are possible. smile
Remember that we are not to considered the ordered pairs here (i.e. 1, 2, 18 and 2, 1, 18 are to be taken as same sets) and further HCF (x, y, z) also need to be 1.

Kamal Lohia 
Re: algebra alert
by soumya sawarn - Saturday, 13 September 2014, 07:09 PM
  Why can't we take x=4,y=1 and z=16 here too the hcf is 1and 1 of the numbers is less than 24
Re: algebra alert
by TG Team - Sunday, 14 September 2014, 12:42 PM
  Hi Soumya smile

(1, 4, 16) is same as that of (4, 1, 16) and has already been counted in case (i). Please check. smile

Kamal Lohia
Re: algebra alert
by soumya sawarn - Sunday, 14 September 2014, 07:35 PM
  Sir I think there is one more case 2,9,10
Re: algebra alert
by TG Team - Monday, 15 September 2014, 11:04 AM
  Yes, you are right Soumya. I missed that. smile

Actually in case (ii), last set should be 2, 9, 10 and previous to that one will be 2, 8, 11 which I've written there wrongly.

Thanks for correction. smile

Kamal Lohia 
Re: algebra alert
by soumya sawarn - Thursday, 18 September 2014, 10:31 PM
  6 black and 6 white balls of the same size are distributed among 10 urns such that each urn contains at least one ball. Find the total number of distributions?
Re: algebra alert
by soumya sawarn - Friday, 19 September 2014, 01:26 AM
  How many four digit numbers contain exactly two distinct digits?
Re: algebra alert
by TG Team - Friday, 19 September 2014, 04:54 PM
  Hi Soumya smile

This is not a CAT-type problem as it involves lot of casework that too with big numbers. You can have an idea for this here. smile

Kamal Lohia
Re: algebra alert
by TG Team - Friday, 19 September 2014, 04:58 PM
  Again a bit of casework. You can go for a similar problem here.

Kamal Lohia
Re: algebra alert
by soumya sawarn - Sunday, 5 October 2014, 11:13 PM
  Two consecutive numbers are removed from a list of first n natural numbers. The average of the remaining numbers is 67/3.what is the product of the two numbers that have been removed?