Six children are each offered a single scoop of any of flavors of ice cream from the Combinatorial Creamery. In how many ways can each child choose a flavor for their scoop of ice cream so that some flavor of ice cream is selected by exactly three children?
There are 3^6 = 729 ways for the 6 children to choose their ice cream flavors if there are no restrictions. However, we need to subtract the number of ways in which no flavor is selected by exactly three children. There are 3 ways to choose the flavor that is selected by exactly three children, and (6 choose 3) ways to choose the three children who will select that flavor. The remaining three children each have two choices for their ice cream flavor. So the total number of ways is 3 * (6 choose 3) * 2^3 = 540.
Therefore, there are 729 - 540 = 189 ways for the 6 children to choose their ice cream flavors so that some flavor of ice cream is selected by exactly three children.
