N = set size
K = size of subsets
X = size of exclusion group

C(A, B) = A! / B!(A-B)!

Then with exclusions

Number of Elements = C(N, K) - C(N-X, K-X) when K-X >= 0
Note this will only work when you are excluding one group of x items

Example 1:
N = 4 (A, B, C, D)
K = 2 (pairs)
X = 2 (AB exlusion)
Number of Elements = C(4, 2) - C(2, 0) = 6 - 1 = 5

Example 2:
N = 4 (A, B, C, D)
K = 3 (Triples)
X = 2 (AB exlusion)
Number of Elements = C(4, 3) - C(2, 1) = 4 - 2 = 2

Example 3:
N = 4 (A, B, C, D)
K = 4 (Quads)
X = 2 (AB exlusion)
Number of Elements = C(4, 4) - C(2, 2) = 1 - 1 = 0