Problem:
Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?
Answer Choices:
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Solution:
If there were no restrictions on the number of candies per bag, then each piece of candy could be distributed in ways. In this case there would be ways to distribute the candy. However, this counts the cases where the red bag or blue bag is empty.
If the red bag remained empty then the candy could be distributed in ways. The same is true for the blue bag. Both totals include the case in which all the candy is put into the white bag. Hence there are ways to distribute the candy such that either the red or blue bag is empty.
The number of ways to distribute the candy, subject to the given conditions, is .
The problems on this page are the property of the MAA's American Mathematics Competitions