If you follow the Service Dog Project, you may see litters of Great Dane puppies

Question

If you follow the Service Dog Project, you may see litters of Great Dane puppies. Suppose there is a litter of 10 puppies. Of these 10, 5 are spotted, 3 are gray, and 2 are black.

(a) Suppose you have been watching this litter attentively enough that you can tell all the puppies apart. How many different ways can you recognize to line up 6 of them at their food bowls?

(b) Now suppose that you can’t tell any of the puppies with the same coat pattern apart (so you can’t tell the gray ones apart, etc.). How many ways are there to line up all 10 of the puppies at their food bowls?

Explanation / Answer

a) In this case, we need to line up 6 out of 10 puppies.

Number of ways it can be done = 10P6 = 10! / (10 - 6)! = 10! / 4! = 151200

b) Now we have to line up 10 puppies in which 5 are of one kind, 3 are of second and 2 are of third kind. Hence,

Number of ways it can be done = 10! / (5! * 3! * 2!) = 2520

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