Problem:
Alina writes the numbers on separate cards, one number per card. She wishes to divide the cards into groups of cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The sum of all the numbers is . This means that the sum of each group is . Consider the group with in it. The other two numbers must add to . Therefore, the other cards in this group are and or and .
Case 1: One group is . Consider the group with in it. The other numbers must add to . The only option is and with the remaining cards. The other group is then , and . This adds to , so this case contributes one possibility.
Case 2: One group is . Consider the group with in it. As above, the other numbers have to add to . The only option is and . The final group is , and , which adds to . This is another configuration. We have gone through all the cases, which revealed that there are only possible groupings.
Thus, is the correct answer.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions