$3$ Squares $3$ Colors

We have the following three possibility:

1. All squares have same color: This can be done be choosing one of the three colors, and painting all the squares with that color. Now, choosing one colour from 3 colours can be done in 3 ways.

2. No two square have same color: For this case, each square has to be given different color. Since the order does not matter, this can be done in 1 way.

3. Exactly two squares have same color: For this we have to first choose two colours and then colours the square. This can be done in $2 \times 3= 6$ . ways.

Thus, total number of ways to colour the squares is $3+1+6 = \boxed{10}$

We have three cases:

case 1 :All squares have same color: This can be done be choosing one of the three colors, and painting all the squares with that color. Now, choosing one colour from 3 colours can be done in 3 ways.

case 2 :No two square have same color: For this case, each square has to be given different color. Since the order does not matter, this can be done in 1 way.

case 3 :Exactly two squares have same color: For this we have to first choose two colours and then colours the square. This can be done in 2*3=6 ways.

Thus, total number of ways to colour the squares is 3+1+6=10

3C1 + 2x 3C2 + 3C3 = 10

(All Squares in the same colors) + (selection of two colors for 3 squares which can be made in two different ways) + (all Squares in different colors) = 10 no. of possibilities

I had to write down all the possibilites - white = w, black = b, red = r (those are the colours that I used). www,bbb,rrr,wbr,wwr,wwb,bbr,bbw,rrw,rrb.