binom52

binom52 is a mathematical and combinatorial concept related to the number of ways to choose 52 items from a set of 52 distinct items. It is a specific case of the binomial coefficient, often denoted as C(52, 52) or "52 choose 52." The binomial coefficient C(n, k) represents the number of combinations of *n* items taken *k* at a time without regard to order.

In combinatorics, C(52, 52) equals 1 because there is exactly one way to select all 52 items from a set of 52 items. This aligns with the general formula for binomial coefficients, where C(n, k) = n! / (k!(n−k)!), and the factorial of 52 (52!) in the numerator and denominator cancels out, leaving 1.

The number 52 is notable in probability and statistics, particularly in card games, where a standard deck consists of 52 cards. While C(52, 52) itself is trivial, related combinations like C(52, 5) (the number of 5-card hands in poker) are widely studied. The symmetry property of binomial coefficients, C(n, k) = C(n, n−k), also applies here, meaning C(52, 52) is equal to C(52, 0), which also equals 1.

binom52 is sometimes referenced in discussions about edge cases in combinatorial mathematics, where trivial selections (either choosing all or none) serve as foundational examples. It reinforces the understanding of how binomial coefficients behave at the extremes of their range.