If all the letters of the word ABOUT are arranged at random in a line, find the odds that the arrangement will start and end with a vowel.?

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answers:
husoski say: Look at it as filling in the blanks: _ _ _ _ _, and figuring out where to put the two consonants. The result will start ad end with a vowel if and only if both consonants (B,T) are in the middle three spots.

There are C(5,2) = 10 ways to pick two spaces for the consonants, but only C(3,2) = 3 ways to put them both in the middle three spots. That's a probability of 3/10 or 0.3 if the letters are randomly arranged.

Check that with a full counting of every permutation. (More work, but it's nice to have another way to get the answer, if only to check for errors.) There are 5! = 120 different ways to rearrange 5 letters, and all of them count because there are no duplicate letters.

Of those, there are C(3,2) = 3 ways to pick two spots in the middle for those consonants, 2! = 2 orders to fill them in and 3! = 6 ways to put the remaining 3 letters in the other spots. That's a total of 3*2*6 = 36 different permutations with both consonants in the middle, meaning vowels at both ends. That's a probability of 36/120, which simplifies to the 3/10 I got above.
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spfor say: mrhntqox
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plnzv say: eqyiouqe
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essbu say: vbxeretj
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