NC1

nC1, read as "n choose 1," denotes the binomial coefficient representing the number of ways to select one element from a set of n distinct elements. In combinatorics this value equals n, since any of the n elements can be chosen and the order of selection does not matter.

Definition and formula: The binomial coefficient nCk is defined for nonnegative integers n and k with 0 ≤ k ≤ n by nCk = n! / (k!(n−k)!). For k = 1, this simplifies to nC1 = n! / ((n−1)!) = n. When n = 0 the value is 0, since there are no elements to choose from.

Relation and applications: nC1 appears as the second entry in the nth row of Pascal's triangle and as the coefficient of x in the expansion of (1 + x)^n, where (1 + x)^n = sum_{k=0}^n nCk x^k. This simple case is fundamental in counting problems, such as selecting a single item from a set of n distinct items, or in probability models involving randomly drawing one element from a population. The concept generalizes to all k between 0 and n and serves as a building block for more advanced combinatorial identities and algebraic expansions.