Determine the number of multisubsets with 12 elements from a

Determine the number of multisubsets with 12 elements from {a, b, c, d} where a occurs at most 4 times, b occurs at most 3 times, c occurs at most 4 times and d occurs at most 5 times.

Solution

Let A = {a,a,a,a,b,b,b,c,c,c,c,d,d,d,d,d}

There are 12 elements in the main set.

The number of subsets would be

when repititions are allowed

= 12c0+12C1+12C2+...+12C12 (including null set and A)

= 2¹²