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Latex binomial coefficient

The binomial coefficient can be interpreted as the number of ways to choose k elements from an n-element set. How to write it in Latex ?

Definition

The binomial coefficient $\binom{n}{k}$ can be interpreted as the number of ways to choose k elements from an n-element set. In latex mode we must use \binom fonction as follows:

\frac{n!}{k!(n - k)!} = \binom{n}{k} = {}^{n}C_{k} = C_{n}^k


$$\frac{n!}{k!(n - k)!} = \binom{n}{k} = {}^{n}C_{k} = C_{n}^k$$

Properties

\frac{n!}{k!(n - k)!} = \binom{n}{k}


$$\frac{n!}{k!(n - k)!} = \binom{n}{k}$$

where A is the permutation

\frac{A_n^k}{k!} = \binom{n}{k}


$$\frac{A_n^k}{k!} = \binom{n}{k}$$

where

A_n^k = \frac{n!}{(n-k)!}


$$A_n^k = \frac{n!}{(n-k)!}$$

are the different ordered arrangements of a k-element subset of an n-set

Pascal’s triangle

\binom{n}{k} =  \binom{n-1}{k-1} +\binom{n-1}{k}


$$\binom{n}{k} = \binom{n-1}{k-1} +\binom{n-1}{k}$$