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 Question from student, a student: Consider all possible arrangements of the word "PROBLEM". If a word is picked at random, find the probability that (a) the word starts with a vowel (b) the word ends with a consonant.

Hi,

The letters of the word problem are distinct so there are 7! different arrangements of the letters. Choose one at random. I am going to find the probability that the middle letter is a vowel.

There are 2 vowels in problem, o and e. If the middle letter is o then there are 6! ways to arrange the other letters and thus there are 6! arrangements of the 7 letter that have o as the middle letter. Likewise there are 6! arrangements of the 7 letter that have e as the middle letter. Hence there are 2 × 6! arrangements of the 7 letter that have o or e as the middle letter. Thus the probability that you choose an arrangement with o or e as the middle letter is (2 × 6!)/(7!) = 2/7.

Harley

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