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### Binomial Coefficients

Blaise Pascal
##### Contents

{$$n \choose k$$} is read "n choose r," the "choose" relating to use in combinations in which it evaluates to the number of ways r objects can be chosen from n objects. An alternate notation is:

{$$_nC_k$$}

#### Definition

{$${n \choose k} = {n! \over {k!(n-r)!}}, \; 0 \le r \le n$$}

#### MathJax Code

 n \choose k OR \binom{n}{k}
OR _nC_k 

The MathJax code generates the symbol including the stretchy parenthesis with the first forms.

#### Values: Pascal’s Triangle

Pascal’s triangle is usually presented in a more pyramidal form, but this tables should make the values for k and n easier to read.

Values of {$_nC_k$} for n and k
n\k 0   1   2   3   4   5   6   7   8   9   10
111
2121
31331
414641
515101051
61615201561
7172135352171
818285670562881
9193684126126843691
101104512021025221012045101

#### Identities

{$$\begin{gather} \binom nk = \frac{n!}{k!\,(n-k)} \cr \binom n0 = \binom nn = 1 \cr {n \choose k} + {n \choose k+1} = {n+1 \choose k+1} \tag{Pascal's Rule} \cr \binom nk = \binom{n-1}{k-1} + \binom{n-1}k \cr \binom{n}{k} = \frac{n}{k} \binom{n-1}{k-1} \cr \binom {n-1}{k} - \binom{n-1}{k-1} = \frac{n-2k}{n} \binom{n}{k} \cr \binom{n}{h}\binom{n-h}{k}=\binom{n}{k}\binom{n-k}{h} \end{gather}$$}

#### Computation

{$$\begin{gather} \binom nk = \frac{n^{\underline{k}}}{k!} = \frac{n(n-1)(n-2)\cdots(n-(k-1))}{k(k-1)(k-2)\cdots 1}=\prod_{i=1}^k \frac{n-(k-i)}{i}=\prod_{i=1}^k \frac{n+1-i}{i} \end{gather}$$}

Sources:

1. File:Blaise Pascal Versailles-cropped.jpg - Wikimedia Commons

Recommended:

Category: Math Algebra

This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong.

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### August 05, 2017

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