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Binomial Coefficients

{$$ n \choose k $$} 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


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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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