OEIS A000332: Binomial Coefficient C(n,4)

A000332 is the binomial coefficient n choose 4 (the number of ways to pick 4 items from n). It is zero for n<4 and equals n(n-1)(n-2)(n-3)/24 for n≥4, giving 1, 5, 15, 35, 70, ... These numbers pop up in geometry, combinatorics, and algebra: for example, the number of interior intersection points formed by diagonals of a convex n-gon (assuming no three diagonals meet at a point); the count of equilateral triangles in a triangular lattice of side length n; the cumulative sums of tetrahedral numbers (linking to the 4-simplex); and the number of independent components of a rank-4 antisymmetric tensor in n dimensions. It also has a neat prime property: 5 is the only prime that appears in this sequence, since for larger n the values are always composite. See how a simple counting formula threads through geometry, higher-dimensional shapes, and even physics.

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