Binomial Coefficient (n Choose k)

The binomial coefficient counts the number of ways to choose k things from n when order does not matter, and it is simultaneously the coefficient of the x^(n−k)y^k term when (x + y)ⁿ is expanded — the two questions are the same question, because each term arises from picking y from k of the n brackets. Worked example: C(10, 3) = (10 × 9 × 8)/(3 × 2 × 1) = 120, so a class of ten yields 120 possible three-person committees, and (x + y)¹⁰ has 120x⁷y³ in it.

Stack these numbers in rows and you get Pascal's triangle, in which each entry is the sum of the two above it. Pascal wrote his Traité du triangle arithmétique in 1654, but the array was already old: Yang Hui printed it in China in 1261 crediting an earlier work of 1100, and Omar Khayyám used it in Persia around the same period. The practical trap is the factorial itself — 52! has 68 digits and overflows any calculator, so never compute the top and bottom separately. Cancel first: C(52, 5) is (52 × 51 × 50 × 49 × 48)/(5 × 4 × 3 × 2 × 1) = 2,598,960, the number of five-card poker hands, and it needs only ten multiplications. Note also the symmetry C(n, k) = C(n, n − k) — choosing 47 cards to discard is the same act as choosing 5 to keep. Both n and k count objects, so enter whole numbers; anything else is rounded to the nearest integer before counting.