letters of the word ‘COMPUTER’

vishal.pathak wrote:In how many ways can the letters of the word ‘COMPUTER’ be arranged such that vowels occupy even position.
[spoiler]If your answer is 4*3*3*2*2*1 then I will want to ask you that why are we not considering the arrangement of consonants. For every arrangement of vowels, we will have 5! arrangements of consonants also
Thoughts??[/spoiler]

Out of 4 even places we need 3 by 4c3 ways
these 3 gets arranged by 3! ways
rest of 5 places can be filled in 5c1 and arranged in 5! ways

hence required no. of arrangements = 4c3*3!*5!=4*3*2*5*4*3*2=2880.

P.S.: regarding your doubt, definitely the arrangement of consonants and vowels both will have to be considered.

Another method to solve this problem can be, cosider 8 blank seats:

_ _ _ _ _ _ _ _

and 8 letters which can sit on those 8 seats: C,O,M,P,U,T,E,R
O,U,E can only sit at even positions: Even position available=4

1st vowel has 4 options
2nd has 3
3rd has 2

Arrangement of vowels=4*3*2

Now consonants are left with 5 positions
1st cons has 5 options
2nd 4
3rd 3
4th 2
5th 1

Arrangement of consonants=5*4*3*2*1

Final arrangement=4*3*2*5*4*3*2=2880

We will definitely consider the arrangement of the consonants!!

Hope this helps