+28 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?
+1768 Each child can choose from 6 flavors of ice cream.
There are (36) ways to choose 3 flavors of ice cream. There are 6 ways to choose the flavor of the first child who gets two scoops of a particular flavor. There are 5 ways to choose the flavor of the second child who gets two scoops of a particular flavor. For the third child, there is only one unique flavor left. There are a total of 6×5×3×(36)=2520 ways for the three children to each have exactly two scoops of a particular flavor.
There are (36) ways to choose 3 flavors of ice cream. There are 6 ways to choose the flavor of the first child who gets one scoop of a particular flavor. For the second and third children, there are 5 ways to choose the flavor of their scoop of ice cream. There are a total of 6×5×5×(36)=3600 ways for the three children to each have exactly one scoop of a particular flavor.
There are (36) ways to choose 3 flavors of ice cream. There are 6 ways to choose the flavor of the first child who gets none of the three flavors. There are 5 ways to choose the flavor of the first child who gets one of the three flavors. There are 4 ways to choose the flavor of the first child who gets two of the three flavors. There are a total of 6×5×4×(36)=1440 ways for the three children to have different flavors.
There is a total of 2520+3600+1440=7560 ways for each child to choose a flavor for their scoop of ice cream so that some flavor of ice cream is selected by exactly three children.
