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A First Course in Probability and Markov Chains by Laura Poggiolini, Giuseppe Modica

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1

Combinatorics

Combinatorics deals with the cardinality of classes of objects. The first example that jumps to our minds is the computation of how many triplets can be drawn from 90 different balls. In this chapter and the next we are going to compute the cardinality of several classes of objects.

1.1   Binomial coefficients

1.1.1   Pascal triangle

Binomial coeffcients are defined as

images

Binomial coefficients are usually grouped in an infinite matrix

images

called a Pascal triangle given the triangular arrangement of the nonzero entries, see Figure 1.1. Here and throughout the book we denote the entries of a matrix (finite or infinite) A = images where the superscript i and the subscript j mean the ith row and the jth column, respectively. Notice that the entries of each row of C are zero if the column index is large enough, images, j with j > i ≥ 0. We also recall the Newton binomial formula,

images

images

Figure 1.1   Pascal ...

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