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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
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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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