On a homework assignment I was given the following:

How many functions are there from {1,...,n} to {1,...,n} that are not 1-to-1?

I immediately thought the answer to this question was zero. I tried proving this with the pigeon hole principal. I said let there be n boxes and let there be n balls. I then have n/n = 1, which tells me that there is one ball in each box which then tells me that there is no function that is not one to one. Are my assumptions correct?