How many (4)-element subsets of (A={1,2,3,4,5,6,7,8}) do not contain both (2) and (3) together?

Correct answer: (55)

The direct answer is option B, 55. There are 8 elements in the set, and we choose 4, so the unrestricted number is C4 = 8!/(4!4!) = 70. The phrase “do not contain both 2 and 3 together” means a subset is forbidden only when it contains both 2 and 3; it may contain neither or exactly one of them. Count the forbidden subsets: if 2 and 3 are already included, choose the remaining 2 elements from the other 6, giving C2 = 15. Hence allowed subsets = 70 − 15 = 55. Option A, 50, subtracts too many or uses a wrong combination. Option B, 55, is correct. Option C, 60, does not remove all forbidden subsets. Option D, 70, counts every subset and therefore includes the 15 invalid ones. A common language mistake is to read the condition as “exclude both separately”; that would be a different question. Memory cue: “not both” means subtract only the cases containing the pair together.