A Probability Course for the Actuaries A Preparation for Exam P/1

44 COUNTING AND COMBINATORICS
(b) P (6, 2) = 30 ways
(c) C(6, 3) = 20 different ways
As an application of combination we have the following theorem which provides
an expansion of (x + y) n , where n is a non-negative integer.
Theorem 4.4 (Binomial Theorem)
Let x and y be variables, and let n be a non-negative integer. Then
(x + y) n =
n∑
C(n, k)x n−k y k
k=0
where C(n, k) will be called the binomial coefficient.
Proof.
The proof is by induction on n.
Basis of induction: For n = 0 we have
(x + y) 0 =
0∑
C(0, k)x 0−k y k = 1.
k=0
Induction hypothesis: Suppose that the theorem is true up to n. That is,
(x + y) n =
n∑
C(n, k)x n−k y k
k=0
Induction step: Let us show that it is still true for n + 1. That is
∑n+1
(x + y) n+1 = C(n + 1, k)x n−k+1 y k .
k=0