What is the formula for combinations?

The formula for combinations is expressed as C(n,k) = n! / [k!(n-k)!]. This formula is essential in combinatorial mathematics, where it is used to determine the number of ways to choose k elements from a set of n elements without regard for the order of selection.

In this formula, n! (n factorial) represents the total number of ways to arrange n elements, while k! accounts for the arrangements of the k elements that are being chosen, which we do not want to count multiple times. Similarly, (n-k)! accounts for the arrangements of the remaining elements that are not selected. This reasoning ensures that we count only unique combinations of elements, rather than permutations, which would consider different orders as distinct.

The other choices do not accurately represent the concept of combinations. For instance, simply adding or subtracting n and k does not pertain to the selection process involved in combinations. Additionally, using only n! or k! does not account for the relationship between the total elements and the chosen elements, failing to provide the necessary context that combinations require. Therefore, the correct formula is indeed that given in the first choice.