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Following are common definition of Binomial Coefficients.

A binomial coefficient C(n, k) can be defined as the coefficient of X^k in the expansion of (1 + X)^n.

A binomial coefficient C(n, k) also gives the number of ways, disregarding order, that k objects can be chosen from among n objects; more formally, the number of k-element subsets (or k-combinations) of an n-element set.