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(combinatorics) a coefficient of any of the terms in the expansion of the binomial (x+y)ⁿ, defined by ${\displaystyle {n \choose k}={\frac {n!}{k!(n-k)!}}}$, read as "n choose k"

Visualisation of binomial expansion up to the 4th power

(combinatorics) a coefficient of any of the terms in the expansion of the binomial (x+y)ⁿ, defined by ${\displaystyle {n \choose k}={\frac {n!}{k!(n-k)!}}}$, read as "n choose k"

1000th row of Pascal's triangle, arranged vertically, with grey-scale representations of decimal digits of the coefficients, right-aligned. The left boundary of the image corresponds roughly to the graph of the logarithm of the binomial coefficients, and illustrates that they form a log-concave sequence.

(combinatorics) a coefficient of any of the terms in the expansion of the binomial (x+y)ⁿ, defined by ${\displaystyle {n \choose k}={\frac {n!}{k!(n-k)!}}}$, read as "n choose k"

Binomial coefficients C (n, k) extended for negative and fractional n, illustrated with a simple binomial. It can be observed that Pascal's triangle is rotated and alternate terms are negated. The case n = −1 gives Grandi's series.