Question

Mr. Smith brings home 7 animals for his 7 children. Each child will adopt a pet to be her or his own. There are 4 different cats (a Siamese, a Persian, a Calico, and a Minx), 2 different dogs (a Poodle and a Golden Retriever), and a goldfish. Anna and Betty refuse to take care of the goldfish, and Charlie and Danny insist on having cats. The other 3 kids are easier to please -- they'll take anything. In how many ways can Mr. Smith give the children pets?

Answer 1

There are 4 cats, 2 dogs and 1 goldfish for a total of 7 animals. These animals should be distributed to each of Mr. Smith's children. But we must note of the restrictions. 

This is the solution for the problem:

2(7C6) + 2(7C4) + 3(7C1)

How is this so? The first term stand for Anna and Betty, hence, 2. Each of them must choose 6 out of the 7 animals (except for goldfish). That's why it's written as '7C6' which is combination of 6 out of 7. It stands for nCr, or 'r' objects out of 'n'. The equation for that is n!/[r!(n-r)!]. The second term is for Charlie and Danny who gets 4 cats out of the 7 animals. And lastly, the other 3 would get the three last pet available.

Solving the equation, the answer is 105 ways.