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 child choose a flavor for their scoop of ice cream so that some flavor of ice cream is selected by exactly three children?

 Apr 13, 2024

We can solve this problem by considering the two unwanted scenarios and subtracting them from the total number of possibilities.

Total number of ways:


There are 3 choices (flavors) for each child, so there are 36=729 total ways for all 6 children to choose their ice cream flavors with no restrictions.


Unwanted Scenario 1: No flavor chosen by exactly 3 children


Choose the flavor that isn't picked by exactly 3: There are 3 ways to choose which flavor will be picked by only 2 or none of the children.


Choose the 3 children who get the non-chosen flavor: There are (36​)=20 ways to choose 3 children out of 6.


Choose the flavors for the remaining 3 children: Each of these children has 2 remaining flavors to choose from, for a total of 23=8 possibilities.


There are 3⋅20⋅8=480​ ways for all flavors to be chosen by either 0, 1, 2, or all 6 children.


Unwanted Scenario 2 (not necessary but can be solved for completeness):


All 3 flavors are chosen by exactly 2 children each. This can be solved similarly to scenario 1, but the calculations are slightly more involved.


Final Answer:


Since we only care about scenarios where exactly one flavor is chosen by 3 children, we subtract the unwanted scenarios from the total:


Total ways - Unwanted Scenario 1 (or Scenario 2) = 729−480 = 249​ ways.

 Apr 13, 2024

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