For questions that explicitly reference the binomial coefficients, Pascal's Triangle, and Binomial identities.
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written as $\dbinom{n}{k}$. It is the coefficient of $x^k$ term in the polynomial expansion of the binomial power $(1+x)^n$; this coefficient can be computed by the multiplicative formula:
$\dbinom{n}{k} = \frac{n(n-1)(n-2)...(n-k+1)}{k(k-1)(k-2)...1}$
which using factorial notation can be compactly expressed as
$\dbinom{n}{k} = \frac{n!}{k!(n-k)!}$
Arranging the numbers $\dbinom{n}{0}, \dbinom{n}{1}, \dbinom{n}{2},..., \dbinom{n}{k}$ gives a triangular array called Pascal's triangle, satisfying the recurrence relation :
$\dbinom{n}{k} = \dbinom{n-1}{k-1} + \dbinom{n-1}{k}$.
The binomial coefficients occur in many areas of mathematics, and especially in combinatorics.
The binomial coefficients can be generalized to $\dbinom{z}{k}$ for any complex number $z$ and integer $k ≥ 0$, and many of their properties continue to hold in this more general form.
