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P & C conceptual doubt
by RUPALI GOYAL - Wednesday, 8 October 2014, 10:29 AM
  Dear Sir,

please help me with the following doubt.

Q. In how many ways the letters in the word MATHEMATICS can be arranged if vowels are to be arranged at extreme ends?

Ans.
Now here, the vowels are A, A, E, I that are to be arranged at ends.
So, these can be arranged in 4P2/2! (using the concept 4 things are to be arranged at 2 places and out of these 2 are same)

Also, rest 9 letters can be arranged in 9!/2! 2! ways.

So, total number of ways will be ( 4P2/2! ) (9!/2! 2!)
My doubt is 4P2/2! will give 6. However, when I write the combination then it comes out to be 7. Please tell me what is the mistake that i have done if I use formula rather than writing the combinations manually.

A...............A,
A...............E,
A...............I,
E...............A,
E...............I,
I................E,
I................A

Regards,
Rupali
Re: P & C conceptual doubt
by Rahul Saxena - Sunday, 12 October 2014, 02:52 AM
  Dear Rupali u have understood the question very well. but there is some implementation prob. look there are 4 vowels to be arranged in 2 places but mind u 2 are simillar, for this you will have to take two cases, firstly on A, E and I is taken and permuted in 2 places, which comes out to be P(3,2)=6 and secondly case when both A's are present (2!)/(2!)=1
so a total of seven cases smile