By Marcel B. Finan

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**Extra info for A Probability Course for the Actuaries: A Preparation for Exam P 1**

**Example text**

There are C(n − n1 − n2 − · · · − nk−1 , nk ) different ways. Thus, applying the Fundamental Principle of Counting we find that the number of distinguishable permutations is n! n2 ! · · · nk ! 48 COUNTING AND COMBINATORICS It follows that the number of different rearrangements of the letters in DE8! = 3360. 3! 1 In how many ways can you arrange 2 red, 3 green, and 5 blue balls in a row? Solution. There are (2+3+5)! 5! 2 How many strings can be made using 4 A’s, 3 B’s, 7 C’s and 1 D? Solution. Applying the previous theorem with n = 4 + 3 + 7 + 1 = 15, n1 = 4, n2 = 3, n3 = 7, and n4 = 1 we find that the total number of different such strings is 15!

The answer is given by n+k−1 k−1 = n+k−1 n It follows that there are C(n + k − 1, k − 1) ways of placing n identical objects into k distinct boxes. 6 How many ways can we place 7 identical balls into 8 separate (but distinguishable) boxes? 5 PERMUTATIONS AND COMBINATIONS WITH INDISTINGUISHABLE OBJECTS51 Solution. 7 An ice cream store sells 21 flavors. Fred goes to the store and buys 5 quarts of ice cream. How many choices does he have? Solution. Fred wants to choose 5 things from 21 things, where order doesn’t matter and repetition is allowed.

How many handshakes took place? 12 There are five members of the math club. In how many ways can the twoperson Social Committee be chosen? 13 A consumer group plans to select 2 televisions from a shipment of 8 to check the picture quality. In how many ways can they choose 2 televisions? 14 A school has 30 teachers. In how many ways can the school choose 3 people to attend a national meeting? 15 Which is usually greater the number of combinations of a set of objects or the number of permutations?