## Quarter Square

#### Problem

A square is split into four smallers squares and exactly two of these smaller squares are shaded. For example, the top left and bottom right squares could be shaded.

How many distinct ways can exactly two smaller square be shaded?

#### Solution

It can be seen that there are exactly six ways:

However, we could have arrived at this answer by a quite different method. Let us consider two distinct cases: the top left square is either shaded or unshaded.

(i) If the top left square is shaded then there are three remaining squares that could be shaded (see the top three diagrams).

(ii) If the top left square is unshaded then there are three remaining squares that could also be left unshaded, which means that the other two must be shaded (see the bottom three diagrams).

Therefore there are $3 + 3 = 6$ ways of shading exactly two smaller square.

We can use this method to explain why there are six ways of picking two numbers from $\{1,2,3,4\}$. We pick $1$ and one other number from $\{2,3,4\}$ or we do not pick $1$ and do not pick one other from $\{2,3,4\}$.

However, when we consider picking two numbers from $\{1,2,3,4,5\}$ we need to adapt the strategy slightly. We pick $1$ and one other number from $\{2,3,4,5\}$: 4 ways, or we do not pick $1$ and do not pick two from $\{2,3,4,5\}$. This second case is equivalent to picking two from $\{1,2,3,4\}$: 6 ways. Hence there are $4 + 6 = 10$ ways of picking two numbers from $\{1,2,3,4,5\}$.

Use the same method to explain why there are 15 ways of picking two from $\{1,2,3,4,5,6\}$.

How many ways can you pick two numbers from $\{1,2,3,4,5,6,7\}$?