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The Problem Solver's Resource

Introduction \| Other Tips and Tricks \| Methods of Proof \| You are currently viewing page 5.

Combinatorics

This section cover combinatorics, and some binomial/multinomial facts.

Permutations

The factorial of a number $n$ is $n(n-1)(n-2)...(1)$ or also as $\prod_{a=0}^{n-1}(n-a)$ ,and is denoted by $n!$ .

Also, $0!=1$ .

The number of ways of arranging $n$ distinct objects in a straight line is $n!$ . This is also known as a permutation, and can be notated $\,_{n}P_{r}$

Combinations

The number of ways of choosing $n$ objects from a set of $r$ objects is $\frac{n!}{r!(n-r)!}$ , which is notated as either $\,_{n}C_{r}$ or $\binom{n}{r}$ . (The latter notation is also known as taking the binomial coefficient.

Binomials and Multinomials

Binomial Theorem: $(x+y)^n=\sum_{r=0}^{n}x^{n-r}y^r$
Multinomial Coefficients: The number of ways of ordering $n$ objects when $r_1$ of them are of one type, $r_2$ of them are of a second type, ... and $r_s$ of them of another type is $\frac{n!}{r_1!r_2!...r_s!}$
Multinomial Theorem: $(x_1+x_2+x_3...+x_s)^n=\sum \frac{n!}{r_1!r_2!...r_s!} x_1+x_2+x_3...+x_s$ . The summation is taken over all sums $\sum_{i=1}^{s}r_i$ so that $\sum_{i=1}^{s}r_i=n$ .

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 ==<span style="font-size:20px; color: blue;">Combinatorics</span>==
 This section cover combinatorics, and some binomial/multinomial facts.

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