Probability and Statistics for Finance

Appendix C. Binomial and Multinomial Coefficients

In this appendix, we explain the concept of the binomial and multinomial coefficients used in discrete probability distributions described in Chapter 9.

Binomial Coefficient

The binomial coefficient is defined as

Equation C.1. 

for some nonnegative integers k and n with 0 ≤ k ≥ n. For the binomial coefficient, we use the factorial operator denoted by the "!" symbol. As explained in Appendix A, a factorial is defined in the set of natural numbers N that is k = 1, 2, 3, ... as

Equation C.2. 

For k = 0, we define 0! ≡ 1.

Derivation of the Binomial Coefficient

In the context of the binomial distribution described in Chapter 9, we form the sum X of n independent and identically distributed Bernoulli random variables Y_i with parameter p or, formally,

Special Case n = 3

We illustrate the special case where n = 3 using a B(3,0.4) random variable X; that is, X is the sum of three independent B(0.4) distributed random variables Y₁, Y₂, and Y₃. All possible values for X are contained in the state space Ω={0, 1, 2, 3}. As we will see, some of these k ∈ Ω can be obtained in different ways.

We start with k = 0. This value can only be obtained when all Y_i are ...