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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?

 Nov 25, 2023
 #1
avatar+1642 
+1

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.

 Nov 25, 2023
 #2
avatar+28 
-1

that seems a little too much, can you recheck it? im not too sure if thats correct, as with no restrections they would all get 6! scoops, or 720.

Icebulb2  Nov 25, 2023
edited by Icebulb2  Nov 25, 2023
 #3
avatar+28 
-1

Yeah, I tried and it wasnt correct. Your answer is too much, it cant work sadly.

Icebulb2  Nov 25, 2023
edited by Icebulb2  Nov 25, 2023
 #4
avatar+118627 
+3

Your question is incomplete.

 Nov 25, 2023

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