Question

Answer the following question using arrangements with repetition, permutations, or combinations. Be sure to explain why the particular counting technique applies to the problem. A dog shelter is giving away 15 different dogs, but you have room for only 5 of them. How many different dog families could you have

Answer 1

Given

In a dog shelter there are 15 different dogs and there is space for only 5 of them.

Explanation

To determine the number of different dogs you could have.

Combinations should be used because no item may be selected more than once and the order does not matter.

$\begin{gathered} 15_{C_5}=\frac{15!}{5!10!} \\ \frac{15!}{5!10!}=\frac{15\times14\times13\times12\times11\times10!}{5\times4\times3\times2\times1\times10!} \end{gathered}$$\begin{gathered} \frac{15\times14\times13\times12\times11}{5\times4\times3\times2\times1}=7\times13\times3\times11 \\ 3003. \end{gathered}$

Answer

Hence the number of different dogs you could have is 3003.