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# 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 ≤ kn. 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 Yi 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 Yi are ...

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