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# Counting question

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We want to divide 10 dogs (including Fluffy and Nipper) into three groups, one with 3 dogs, one with 5 dogs, and one with 2 dogs.  How many ways can we form the groups such that Fluffy is in the 2-dog group and Nipper is in the 5-dog group?

Jan 13, 2023

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There are $${8 \choose 4} = 70$$ ways to choose dogs for the group of 5.

There are $${4 \choose 3} = 4$$ ways to choose the dogs for the group of 3.

Then there is $${1 \choose 1} = 1$$ ways to choose the dogs for the group of 1.

So, there are $$70 \times 4 \times 1 = \color{brown}\boxed{280}$$

Jan 13, 2023