### Instruction: Alex has 5 pieces of paper, each with a different math problem. In how many ways can he give these problems to his 10 friends (he can give each friend more than one problem)?### Response: This is a problem of counting the number of functions from a set of size 5 (the problems) to a set of size 10 (the friends).
One way to think about it is to imagine assigning each problem to a friend by writing the friend's name next to the problem.
For the first problem, I have 10 choices of friends to write next to it.
For the second problem, I still have 10 choices of friends, since I can give it to the same friend as the first problem or a different one.
Similarly, for each of the remaining problems, I have 10 choices of friends.
So, by the multiplication principle, the total number of ways to assign the problems is 10 * 10 * 10 * 10 * 10, or 10^5.
Another way to think about it is to use the formula for the number of functions from a set of size m to a set of size n, which is n^m.
Here, m is 5 and n is 10, so the formula gives 10^5 as well.