Six children are each offered a single scoop of any of $3$ flavors of ice cream from the Combinatorial Creamery. In how many ways can each c

There are two ways to solve this problem.

 

Method 1: Using complementary counting

We can solve this problem by finding the number of ways in which no flavor is selected by exactly three children and subtracting from the total number of ways in which the children can choose their ice cream flavors.

The total number of ways in which the children can choose their ice cream flavors is 36=729.

There are three ways to choose which flavor will be selected by exactly three children. Once we have chosen the flavor, we can assign it to any three children in (36​)=20 ways.

The remaining three children will each have 2 choices for their ice cream flavor. So the total number of ways in which no flavor is selected by exactly three children is 3×20×23=540.

Therefore, the number of ways in which some flavor of ice cream is selected by exactly three children is 729−540=189​.

 

Method 2: Using casework

We can also solve this problem by considering the three cases where the flavor selected by exactly three children is:

The chocolate flavor.

The vanilla flavor.

The strawberry flavor.

In each case, we can count the number of ways to choose the three children who will get that flavor and the number of ways to choose the flavors for the remaining three children.

The total number of ways is then the sum of the number of ways in each case.

The casework is as follows:

Case 1: The chocolate flavor.

There are (36​)=20 ways to choose the three children who will get chocolate. The remaining three children can each choose from the two remaining flavors, so there are 23=8 ways to choose their flavors.

Case 2: The vanilla flavor.

The number of ways is the same as in Case 1.

Case 3: The strawberry flavor.

The number of ways is the same as in Case 1.

Therefore, the total number of ways is 20×8+20×8+20×8=189.