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What is the shortest and most systematic way of solving the below question?

In how many ways can the letters of the word "COMPUTER" be arranged if vowels occupy the even positions?

regards, Jack

I think it's meant that vowels can occupy only even positions: C|O|M|P|U|T|E|R O|E|O|E|O|E|O|E (O and E stand for odd and even positions respectively).

# of arrangements would be \(C^3_4*3!*5!=4!*5!=2880\).

\(C^3_4\) - choosing which 3 even positions out of 4 will be occupied by vowels (there are 4 even positions: 2nd, 4th, 6th and 8th and only 3 vowels); \(3!\) - # of different arrangements of these vowels on their even positions; \(5!\) - # of different arrangements of 8-3=5 other letters left.

The word COMPUTER consists of 8 letters of which 3 vowels. All letters are distinc. First, put 3 vowels into 3 of the positions 2, 4, 6, and 8. There are 4*3*2=24 ways. Second, once the positions of the 3 vowels are determined, we have to put the 5 other letters into the remaining positions. There are 5*4*3*2*1=120 ways. Therefore, there are a total of 120*24=2880 ways.
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Re: In how many ways can the letters of the word "COMPUTER" [#permalink]

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23 Sep 2017, 22:25

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