What identity is true regarding combinations in permutations?

In the context of combinatorial mathematics, the identity that holds true regarding combinations is known as the symmetry property of combinations. Specifically, the correct relationship is that the number of combinations of selecting ( r ) items from a set of ( n ) items, denoted as ( nCr ), is equal to the number of combinations of selecting ( n - r ) items from the same set. This is expressed mathematically as ( nCr = nC(n-r) ).

This identity stems from the combinatorial reasoning that choosing ( r ) items to include in a group is directly related to choosing ( (n - r) ) items to exclude from the same group. Therefore, both selections yield the same number of combinations, affirming the equality stated in the correct option.

In practical terms, if you have a total of ( n ) items and you want to find out how many ways you can choose ( r ) of them, you are essentially determining how many ways you can exclude ( (n - r) ) items from those ( n ) items, leading to the same result, thus confirming this identity as valid and robust in combinatorial theory.