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→Combinations: add the binomial formula, a canonical application of binomial coefficients
:<math>\begin{pmatrix} n \\ r \end{pmatrix} = \frac{n!}{(n-r)!r!}</math>
:<math>\begin{pmatrix} n \\ r \end{pmatrix} = \frac{n!}{(n-r)!r!}</math>
For example, take a race which has 10 competitors. There are 10 choose 3 different combinations of medal winners (that is, racers who get either 1st, 2nd, or 3rd place), which is a total of 10! / (10 - 3)! • 3! = 10! / 7! • 3! = 120 different combinations.
For example, take a race which has 10 competitors. There are 10 choose 3 different combinations of medal winners (that is, racers who get either 1st, 2nd, or 3rd place), which is a total of 10! / (10 - 3)! • 3! = 10! / 7! • 3! = 120 different combinations.
The numbers <math>\begin{pmatrix} n \\ r \end{pmatrix}</math> are also called binomial coefficients because of the expansion:
:<math>(x + y)^n = \sum_{r = 0}^n \begin{pmatrix} n \\ r \end{pmatrix} x^r y^{n-r}</math>
==References==
==References==