Permutations/Combinations

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Permutations/Combinations [#permalink]

23 Oct 2009, 19:34

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On how many ways can the letters of the word "COMPUTER" be arranged?
1. M must always occur at the third place
2. Vowels occupy the even positions.

Ans 1)

[Reveal] Spoiler:

5040 or 7P2*5P2

Ans 2)

[Reveal] Spoiler:

4 * 720

Someone please explain 2, I understand 1
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Re: Permutations/Combinations [#permalink]

26 Oct 2009, 21:58

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#2 doesn't make any sense. It says "vowels have to occupy the even positions" but there are not enough vowels to occupy the even positions. This is where the 4 comes in.

First, if you assume that the vowels occupy slot numbers 2, 4 and 6, then that leaves slot 8 open for a consonant.

The consonant's can be arranged in 5! ways, and the vowels in 3! ways. Together, that would be 120 * 6 = 720. But, that is assuming that the vowels occupy only slot #2, 4 and 6. Now, those vowels can be arranged in 3 of the 4 available slots. This is the same as selected 3 out of 4, which is also the same as selecting 4 (remember that in selecting 3, you're selecting 1 that is left out) This means that for any combination of placement for vowels, multiply that number by 4 and you get the number total.

Example: Vowels = O, U, E

2=0, 4=U, 6=E 8={Non-vowel}
In the same order of OUE, you could have 2={non vowel} 4=O 6=U 8=E. Same order, but different combination possibility.

So 4 * 720 accounts for the fact that vowels can be placed in any 3 of the 4 open "even" spots. Hope this helps.

srini123 wrote:

On how many ways can the letters of the word "COMPUTER" be arranged?
1. M must always occur at the third place
2. Vowels occupy the even positions.

Ans 1)

[Reveal] Spoiler:

5040 or 7P2*5P2

Ans 2)

[Reveal] Spoiler:

4 * 720

Someone please explain 2, I understand 1

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J Allen Morris
**I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$.

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Senior Manager

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Re: Permutations/Combinations [#permalink]

27 Oct 2009, 06:26

thanks Jallen , that makes sense +1 Kudos
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Re: Permutations/Combinations [#permalink]

27 Oct 2009, 10:43

I know this has already been answered but this seemed like an easier solution.
First we find out the number of ways 3 vowels (e,o,u) occupy even positions 2,4,6,8.
this is picking 3 positions out of the 4 where order is imp hence
4P3
the other 5 consonants can occupy any any of the remaining 5 positions hence
!5

Answer is !4*!5 = 24*120 = 4*720.
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Re: Permutations/Combinations [#permalink]

27 Oct 2009, 10:48

It's faster only in that I went to great lengths (as I usually do) to make sure we understand WHY we must answer a problem in a certain way. This way, when someone gets a similar problem, but not identical, that person has confidence and can answer that problem too. Guess it's just the teacher in me.

atish wrote:

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Re: Permutations/Combinations [#permalink]

30 Nov 2009, 20:44

Thank you both... thats a lot of help.
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-------------------------------
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Re: Permutations/Combinations [#permalink] 30 Nov 2009, 20:44

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