By Marcel B. Finan

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

**Example text**

Because there are still C(5, 2) = 10 possible ways to choose the 2 women, it follows that there are 30 · 10 = 300 possible committees The next theorem discusses some of the properties of combinations. 3 Suppose that n and k are whole numbers with 0 ≤ k ≤ n. Then (a) C(n, 0) = C(n, n) = 1 and C(n, 1) = C(n, n − 1) = n. (b) Symmetry property: C(n, k) = C(n, n − k). (c) Pascal’s identity: C(n + 1, k) = C(n, k − 1) + C(n, k). 4 PERMUTATIONS AND COMBINATIONS 41 Proof. n! (n−0)! n! n! n! = 1. (n−1)! = n and C(n, n − 1) = (n−1)!

For such problems the counting of the outcomes is simplified by means of algebraic formulas. 1 If a choice consists of k steps, of which the first can be made in n1 ways, for each of these the second can be made in n2 ways,· · · , and for each of these the kth can be made in nk ways, then the whole choice can be made in n1 · n2 · · · · nk ways. Proof. In set-theoretic term, we let Si denote the set of outcomes for the ith task, i = 1, 2, · · · , k. Note that n(Si ) = ni . Then the set of outcomes for the entire job is the Cartesian product S1 × S2 × · · · × Sk = {(s1 , s2 , · · · , sk ) : si ∈ Si , 1 ≤ i ≤ k}.

11 At the beginning of the second quarter of a mathematics class for elementary school teachers, each of the class’s 25 students shook hands with each of the other students exactly once. 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.

### A Probability Course for the Actuaries: A Preparation for Exam P 1 by Marcel B. Finan

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