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

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.

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,

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 *Y*_{1}, *Y*_{2}, and *Y*_{3}. 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 ...

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