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I'm reading Lady Luck by Warren Weaver. My answers for two of the problems in chapter 4 don't match the given answers.

  1. How many three-letter words, each to contain one of the five regular vowels, can be made from an alphabet of 26 letters, permitting repeititions?

given answer: (5)(26)(25) + (26)(5)(26) + (26)(26)(5) my answer: (5)(21)(21) + (21)(5)(21) + (21)(21)(5) If the vowel is in position 1, there are 5 choices. For each of those choices, there are 21 consonant choices (26-5=21) for the second position and 21 consonant choices for the third position. Same reasoning and number for words with the vowel in the second position and third position.

Since repetition of consonants is allowed, the 25 in the first factor of the book's given answer makes no sense to me. The factors of 26 in the book-answer's second and third terms also seem wrong to me since the problem states each word contains only one vowel.

  1. How many types of banana splits can Howard Johnson's make with any two of twenty-eight sorts of ice cream and anyone of ten syrup flavors?

given answer: 7560 my answer: (28 choose 2)(10) = (28)(27)/2 = 3780. There are 28 ways to choose first ice cream flavor. for each of those ways, there are 27 ways to choose the second flavor. However, this includes duplicates like chocolate vanilla and vanilla chocolate so have to divide by number of ways to arrange 2. So that part is 28 choose 2. For each of those 28 choose 2 ice cream combinations, there are 10 syrup flavors to choose from. So 28 choose 2 times 10 = 3780. I think the book-answer over-counts by a factor of 2.

Can someone explain why my reasoning is incorrect? Or do I have the correct answers and the book-given answers are wrong.

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When they say the words can be made with repetitions, you can also repeat the vowels. However the book's answer is also wrong.

Consider that words with at least one vowel = all words - words with no vowels

and words with no vowels = $21^3$

Thus the number of words you want is $26^3 - 21^3 = 8315$.

The book over-counts some words: the book enforces a position to be a vowel and then admits repetitions on the other positions, which is fine, but then over-counts the number of words that have 2 or more vowels.

For your second question, it surely has to do with the fact that a chocolate-vanilla banana split is not the same as a vanilla-chocolate one. Maybe they put the ice-cream on top of each other and then the syrup is poured only on top of one? Don't know, never really seen a banana split.

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