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Binomial Coefficient

Middle School

Definition

The binomial coefficient (n k) or ⁿCₖ is the number of ways to choose k items from n items. Also coefficients in binomial expansion.

\binomnk = (n!)/(k!(n-k)!)

Definition of binomial coefficient

\binomnk = \binomnn-k

Symmetry

\binomnk + \binomnk+1 = \binomn+1k+1

Pascal's identity

(x+y)ⁿ = ∑_k=0ⁿ \binomnk x^n-k y^k

Binomial theorem

Example 1

How many ways to choose 3 from 10 people?

Example 2

Expand (x+y)³.

Discovered by: Blaise Pascal (1654)

Systematized with Pascal's triangle, but known since ancient times.

Binomial distribution

Counting

Sample selection