how many ways to arrange

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ket: Senior | Next Rank: 100 Posts; Posts: 93; Joined: Sun Mar 15, 2009 9:07 am; Thanked: 7 times

by ket » Thu Jun 11, 2009 3:58 am

How many ways are there to rearrange the letters in the word 'elation', if the first and the last letter mus be each a vowel?

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Source: Beat The GMAT — Problem Solving | Thu Jun 11, 2009 3:58 am

rah_pandey: Master | Next Rank: 500 Posts; Posts: 122; Joined: Fri May 22, 2009 10:38 pm; Thanked: 8 times; GMAT Score:700

by rah_pandey » Thu Jun 11, 2009 4:20 am

No of vowels =4
no of ways of choosing vowels=4c2 ways
now you can interchange the two vowels therefore
total no of ways=4c2*2!=4P2=12

no of ways to arrange remaining words=5!

ans=1440=5!*12

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ket: Senior | Next Rank: 100 Posts; Posts: 93; Joined: Sun Mar 15, 2009 9:07 am; Thanked: 7 times

by ket » Thu Jun 11, 2009 4:27 am

that's correct rah_pandey.
I also solved it after I posted it:) But thought would be interesting to others maybe...

Though in my approach instead of this:
no of ways of choosing vowels=4c2 ways
now you can interchange the two vowels therefore
total no of ways=4c2*2!=4P2=12

I simply used the formula n!/[(n-k)!] which counts number of ways things can be arranged when order matters. then I multiplied it by 5! just like you, thus 1440

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dtweah: Master | Next Rank: 500 Posts; Posts: 487; Joined: Fri Mar 27, 2009 5:49 am; Thanked: 36 times

Re: how many ways to arrange

by dtweah » Thu Jun 11, 2009 4:30 am

ket wrote:How many ways are there to rearrange the letters in the word 'elation', if the first and the last letter mus be each a vowel?

For the first letter you have 4 choices and for the last you have 3. So one example is

elatino

But if you reverse the position of the first and last you have a different arrangement:

olatine

After your first and last are decided, you will have 2 letters gone from 7 so only 5 letters to choose from for your middle letters. This is just 5! Putting all together we have:

4 x 5! x3 =1440

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