1996 AJHSME Problems/Problem 21
Problem
How many subsets containing three different numbers can be selected from the set so that the sum of the three numbers is even?
Solution
To have an even sum with three numbers, we must add either , or , where represents an odd number, and represents an even number.
Since there are not three even numbers in the given set, is impossible. Thus, we must choose two odd numbers, and one even number.
There are choices for the even number.
There are choices for the first odd number. There are choices for the last odd number. But the order of picking these numbers doesn't matter, so this overcounts the pairs of odd numbers by a factor of . Thus, we have choices for a pair of odd numbers.
In total, there are choices for an even number, and choices for the odd numbers, giving a total of possible choices for a 3-element set that has an even sum. This is option .
See Also
1996 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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