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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?

Jul 22, 2024

#1
+1790
+1

You are missing a value.

a single scoop of any of  flavors of ice cream

How many flavors of ice cream.

Jul 22, 2024
edited by NotThatSmart  Jul 22, 2024
#2
+19
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So sorry! It's

"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?"

helpmeeee  Jul 22, 2024
#3
+1790
+1

OK, sure! no problem. I'm not 100% sure of my answer, but I'll give it my best.

Let's use case work to solve this question. I'm assuming that children cannot choose to not have ice cream.

First, let's say 3 children chose flavor A.

There are $$\binom 63 ​\cdot 2^ 3 =20⋅8=160$$

This is because we need to choose 3 out of the 6 children to have one flavor, and the other 3 can choose one of the other 2.

The same applies to flavors B and C.

Thus, we have $$160 \cdot 3 = 480$$

I THINK 480 is the answer. Disclaimer, my counting skills are sub-level...but yeah...

Thanks! :)

NotThatSmart  Jul 22, 2024
edited by NotThatSmart  Jul 22, 2024