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By C. Vasudev

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Solution. The word committee contains letters c, o and i once and m, t and e twice. When a word is formed from these letters, a letter may appear at the most the number of time it appear in the word committee or not at all. So generating function for c, o and i is given by (1 + x) each, whereas, for c, m and e is given by ⎛ x2 ⎞ ⎜⎜ 1 + x + ⎟ each. 2 ! ⎠ ⎝ 3 ⎛ x2 ⎞ 1 + x + ⎜⎜ ⎟ 2 ! ⎟⎠ ⎝ 3 If words are to be formed taking all the letters at once, then the numbers of such words is given by the coefficient of x9 9!

The number of different types of books is 10 + 10 = 20. A gift coupon recipient may select any 6 books, possible 6 copies of a single book. This is a case of selection of 6 objects from 20 objects with repetitions allowed. The number of ways the selection can be made is 20 + 6 – 1 C6 = 25C6 = 177100. 90. In an election the number of candidates is one more than the number of vacancies. If a voter can vote in 30 different ways, find the number of candidates. Solution. Let the number of candidates be x.

71. Compute the number of distinct five-card hands than can be dealt from a deck of 52 cards. Solution. This number is 52C5 because the order in which the cards were dealt is irrelevant. 52C5 = 52 ! 5 ! 47 ! or 2,598,960. 72. In how many ways can a prize winner choose three CDs from the top ten list if repeats are allowed ? Solution. Here n is 10 and k is 3. There are (10 + 3 – 1)C3 or 12C3 ways to make the selections. The prize winner can make the selection in 220 ways. 73. How many different seven-person committees can be formed each containing three women from an available set of 20 women and four men from an available set of 30 men ?

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