Binomial coefficient

The binomial coefficients are defined by the formula $${}_nC_k:={n \choose k} = \dfrac{n!}{(n-k)!k!}.$$

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  Graph of the complex binomial coefficient function.

Properties

Proposition: $\displaystyle{n \choose k} = {n \choose {n-k}} = (-1)^k {{k-n-1} \choose k}$

Proof: █

Proposition: $\displaystyle{{n+1} \choose k} = {n \choose k} + {n \choose {k-1}}$

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Proposition: ${n \choose 0} = {n \choose n} = 1$

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Proposition: $1 + \displaystyle {n \choose 1} + {n \choose 2} + \ldots + {n \choose n} = 2^n$

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Proposition: $1 - \displaystyle {n \choose 1} + {n \choose 2} - \ldots + (-1)^n {n \choose n} =0$

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Theorem (Binomial Theorem): $(a+b)^n = \displaystyle\sum_{k=0}^n {n \choose k} a^k b^{n-k}$

Proof: █

Videos

Pascal's Triangle and the Binomial Coefficients Example of choose function (Binomial Coefficient)

References

Abramowitz and Stegun
The Binomial Coefficient Function Binomial coefficients