Problem:
How many subsets containing three different numbers can be selected from the set
so that the sum of the three numbers is even?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The sum of three numbers is even if all three numbers are even, or if two numbers are odd and one is even. Since there are only two even numbers in the set, it follows that the three numbers must include two odd numbers and one even. The possibilities are:
Thus there are possibilities.
Let stand for odd and stand for even. The numbers given are and . There are only two ways that the sum of three numbers is even.
Case I: , and
Case II: .
Since there are only two even numbers, Case I cannot happen. For Case II, , counting choices yields choices for the first odd, remaining choices for the second odd, and choices for the even, for a total of 24 choices. However, since , the number of choices is reduced by a factor of . Hence there are choices.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions