A problem on counting the number of elements in a set consisting of all those matrices which can be multiplied with 5 by 5 permutation matrix

Question

S={A:AB,where B is 5 by 5 permutation matrix}

The answer is given to be 5!=120.

I was thinking A can be n by 5 matrix where n is any natural number,so S should have infinitely many elements.

Is the question incomplete or what I have thought is wrong?

Here is the question if whatever I typed is not clear

Answer 1

In your attachment, $S$ is indeed the set of all $5\times5$ permutation matrices, contrary to the definition of $S$ that you typed. There are $n!$ many $n\times n$ permutation matrices, so indeed $5!$ many $5\times5$ permutation matrices. As a hint for counting: Ask yourself, how many possibilities are there for the first column (or row) of a $5\times5$ permutation matrix? How many for the second, given a choice of the first column (or row)?