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

Binomial theorem
Binomial coefficient (n choose k) equals (n choose (n-k))
Binomial coefficient (n choose k) equals (-1)^k ((k-n-1) choose k)
Binomial coefficient ((n+1) choose k) equals (n choose k) + (n choose (k-1))
Binomial coefficient (n choose 0) equals 1
Binomial coefficient (n choose n) equals 1
Sum over bottom of binomial coefficient with top fixed equals 2^n
Alternating sum over bottom of binomial coefficient with top fixed equals 0

Videos

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

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 3.1.2
• Abramowitz and Stegun
  
• The Binomial Coefficient Function