Find one binomial coefficient equal to the following...

Find one binomial coefficient equal to the following expression:
$$
\left(\begin{array}{l}
n \\
k
\end{array}\right)+3\left(\begin{array}{c}
n \\
k-1
\end{array}\right)+3\left(\begin{array}{c}
n \\
k-2
\end{array}\right)+\left(\begin{array}{c}
n \\
k-3
\end{array}\right) .
$$

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Key Concepts

Binomial Coefficient

A binomial coefficient, typically denoted as C(n, k) or (n choose k), represents the number of ways to choose k elements from a set of n distinct elements without regard to order. It is defined by the formula C(n, k) = n! / (k!(n-k)!) and plays a central role in combinatorics, probability, and algebra, particularly in the expansion of binomials.

Vandermonde's Identity

Vandermonde’s Identity is a combinatorial identity that relates the sum of products of binomial coefficients to a single binomial coefficient. Specifically, it states that for nonnegative integers m, n, and k, the sum ?[i=0 to k] C(m, i) * C(n, k-i) = C(m+n, k). This identity is extremely useful in simplifying sums involving binomial coefficients and underlies many combinatorial proofs and algebraic manipulations.

Convolution of Binomial Coefficients

The convolution of binomial coefficients involves combining sequences by summing products of coefficients from these sequences, usually with weights derived from another binomial expansion. In many cases, such as in the provided expression, the pattern of coefficients (1, 3, 3, 1) arises from a binomial expansion, and convolving these with another sequence of binomial coefficients allows one to sum the series into a single binomial coefficient, revealing deeper combinatorial interpretations.

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