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Binomial coefficients C(n,k) = n!/(k!(n-k)!) count the ways to choose k items from n items. These coefficients appear as entries in Pascal's triangle and satisfy the recursive property C(n,k) = C(n-1,k-1) + C(n-1,k). They also form the coefficients in the expansion of (a+b)n.
You already know from combinations and selections that C(n,k) = n!/(k!(n-k)!) counts the number of ways to choose k items from n without regard to order. Binomial coefficients are exactly these counting numbers — the same formula, now given a geometric home in Pascal's triangle. Pascal's triangle arranges these values in a triangular grid: row n contains C(n,0), C(n,1), ..., C(n,n). The outermost entries are always 1 — there's exactly one way to choose nothing, and one way to choose everything — and every interior entry equals the sum of the two entries directly above it.
That addition rule has a beautiful combinatorial explanation, known as Pascal's identity: C(n,k) = C(n-1,k-1) + C(n-1,k). Imagine you have n objects and want to choose k. Pick one special object — call it "the red ball." Every size-k selection either includes the red ball or it does not. If it includes the red ball, you're choosing the remaining k-1 from the other n-1 objects: C(n-1,k-1) ways. If it excludes the red ball, you're choosing all k from the other n-1 objects: C(n-1,k) ways. These two cases are mutually exclusive and exhaustive, so their sum equals C(n,k). The triangle's rule is not just arithmetic — it encodes this logical split.
The most important application of binomial coefficients is the binomial theorem: (a+b)n = Σ C(n,k) an-k bk. To see why, expand (a+b)n as the product of n copies of (a+b). Each term in the full expansion comes from picking either a or b from each factor. The coefficient of an-k bk is exactly the number of ways to choose which k factors contribute b — which is C(n,k). The triangle's rows literally encode the expansion coefficients: row 2 gives 1, 2, 1 for (a+b)², row 3 gives 1, 3, 3, 1 for (a+b)³, and so on.
Pascal's triangle also harbors deeper patterns. The entries in each row sum to 2n (the number of all subsets of an n-element set). The hockey stick identity — that summing a diagonal gives the entry one row below — corresponds to a counting argument about choosing from sets of growing size. These patterns are not coincidences; they each reflect a combinatorial identity provable by the same logic used for Pascal's identity. Binomial coefficients are the connective tissue between combinatorics, algebra, and probability, and recognizing them in new settings is a core skill in discrete mathematics.