15C3

15C3, read as “fifteen choose three,” is a binomial coefficient that counts the number of ways to select three distinct elements from a set of fifteen when order does not matter. It is a fundamental quantity in combinatorics and probability.

It is calculated by the formula 15!/(3!12!), which simplifies to (15×14×13)/(3×2×1) = 455. This value can also be obtained from the ratio of permutations: 15P3 = 15×14×13 = 2730, and since 15C3 = 15P3/3!, we get 2730/6 = 455. Notation for this quantity includes 15C3, C(15,3), or nCr with n = 15 and r = 3.

In the context of the binomial theorem, 15C3 is the coefficient of the term x^3 in the expansion of (1 + x)^15. It also appears in Pascal’s triangle in row 15, where the third entry from the left (counting from zero) is 15C3.

Applications of 15C3 include counting 3-element subsets from a 15-element set, such as forming 3-person committees, selecting 3 items from a collection of fifteen without regard to order, or determining the number of different 3-card hands from a deck with a chosen subset of fifteen distinct cards.

Related concepts include 15P3 = 2730, the number of ordered 3-element permutations, with 15C3 = 15P3/3! illustrating the relationship between combinations and permutations. Also, 15C3 equals 15C12, reflecting the symmetry property C(n,r) = C(n,n−r).