Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set A × B, each having at least three elements is :

The set A × B will have 4 * 2 = 8 elements.

The number of subsets of a set with n elements is 2^n. So, the number of subsets of the set A × B is 2^8 = 256.

However, this includes the empty set and all subsets with only one or two elements. We need to subtract these from the total.

The number of subsets with one element is simply the number of elements in the set, which is 8.

The number of subsets with two elements is given by the combination formula nCk = n! / [k!(n-k)!], where n is the number of elements in the set and k is the number of elements in the subset. So, the number of subsets with two elements is 8C2 = 8! / [2!(8-2)!] = 28.

So, the number of subsets of the set A × B, each having at least three elements is 256 - 1 (for the empty set) - 8 (for the subsets with one element) - 28 (for the subsets with two elements) = 219.