If A = {x : x ≤ 10 and x ∈ N}, B = {2} and C = {7} and S is a subset of A such that n(S ∩ B) = 1 and n(S ∩ C) = 0, how many such subsets are possible?

The problem is asking for the number of subsets, S, of set A that intersect with set B in exactly one element and do not intersect with set C at all.

Step 1: Define the sets Set A = {x : x ≤ 10 and x ∈ N} is the set of natural numbers less than or equal to 10. So, A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Set B = {2} and Set C = {7}

Step 2: Determine the conditions for set S S is a subset of A such that it intersects with B in exactly one element and does not intersect with C at all. This means that S must contain the number 2 (from set B) and cannot contain the number 7 (from set C).

Step 3: Determine the possible subsets The remaining elements of set A that can be included in set S are {1, 3, 4, 5, 6, 8, 9, 10}. There are 8 elements, so the number of possible subsets is 2^8 (as each element can either be included or not included in the subset).

Therefore, there are 256 possible subsets, S, that satisfy the given conditions.