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Six children are each offered a single scoop of any of 3 flavors of ice cream from the Combinations Creamery. 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 21, 2023
 #1
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There are 372 ways.

 Apr 21, 2023
 #2
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Ways to be all the same flavour = 3

Ways exactly 5 can be the same = 6C5*3*2 = 36

Ways exactly 4 can be the same = 6C4*3*2*2 = 180

Ways exactly 3 can be the same  = 6C3*3*2*2*2 = 480

 

Add them up = 699

 

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another thought...

 

How many ways altogther, no conditions   3^6 = 729

 

There are no ways to not double up the flavour, I mean they cannot all be differerent.

The only way  NOT to have 3 or more of one flavour is for there to be 2 of each flavour.

6!/(2*2*2) = 90

 

 

729-90 = 639.

 

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If my logic was correct these 2 answers would be the same.  

They are not the same so my logic is flawed.

 

MAYBE SOMEONE CAN POINT OUT THE LOGIC ERROR THAT I HAVE MADE.    frown

 Apr 22, 2023

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