How many two-element subsets of{1,2,3,4} are there that do not contain the pair of elements 2 and 4 ?
a) one b) two c)four d) five e) six
a) one b) two c)four d) five e) six
One's initial inclination when one is doing this question might be to remove 2 and 4 from the set, {1, 2, 3, 4}, and say that the remaining two elements, 1 and 3, constitute the one two element subset that can be created.
However, that such an easy answer would be the right one does not make sense, and in fact, what the question is actually asking is how many subsets can be created that do not contain within them BOTH 2 and 4. So any subset that contains either 2 or 4 works.
Using 4 elements chosen 2 at a time we can create (4 x 3)/(2 x 1) = 6 subsets.
Of those two element subsets, the only one that includes both 2 and 4 is the subset {2, 4}.
So the number of two element subsets that don't include the pair 2 and 4 is 6 - 1 = 5.
The correct answer is D.
