nxn square within an mxm square

Question

How would we find the number of ways of placing an nxn square within an mxm square? For example, say we had a 3x3 square. There are 4 ways of placing a 2x2 square within this 3x3 square. Likewise, there are 784 ways of placing a 5x5 square within a 32x32 square. Please see the image below for what exactly is meant by a 'square'.

Answer 1

Consider first fixing the movement to one particular row. You can move $m-(n-1) = m -n + 1$ ways. The same argument holds for the number of columns: $m - n + 1$. You then multiply the two together to get the resulting:

$$\begin{align} (m - n + 1)(m - n + 1) &= m^2 - mn + m - nm + n^2 - n + m - n + 1 \\ &= m^2 - 2nm + m + n^2 - n + m - n + 1 \\ &= m^2 - 2nm + 2m - 2n + n^2 + 1 \end{align}$$

Answer 2

A basic principle of combinatorics is the following: If you can choose independently an element of some $m$ element set and an element of some $n$ element set there are $N=m\,n$ choices in total.