Probability An Introduction by Samuel Goldberg

Author:Samuel Goldberg
Language: eng
Format: epub
Publisher: Dover Publications
Published: 1991-06-26T16:00:00+00:00

2. Binomial coefficients

The numbers introduced in the preceding section have many interesting and important properties that we will need to know for our later work. Although it is a slight digression at this point, we pause here to develop some of these properties.

Our first task is to explain the title of this section. The reader is familiar with the formulas

For any positive integer n, (x + y)n is the product of n equal factors:

In seeking a general formula for (x + y)n, it is helpful first to identify the coefficients in the special cases (2.1)–(2.3). Indeed, the reader should check that these can be written as follows, thus suggesting our first theorem:

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