Problem:
Let set be a -element subset of , and let be the sum of the elements of . Find the number of possible values of .
Solution:
The least possible value of is , and the greatest possible value is . Furthermore, every integer between and , inclusive, is a possible value of . To see this, let be a -element subset the sum of whose elements is , and let be the smallest element of such that is not an element of . Because , conclude that . Therefore, for every -element subset with sum , where , a -element subset with sum can be obtained by replacing by . Thus there are possible values of .
The problems on this page are the property of the MAA's American Mathematics Competitions