Question

A square is bisected vertically and horizontally into 4 smaller squares, and each of the 4 smaller squares is to be painted so that adjacent squares have dif- ferent colors. if there are 20 paints available, in how many ways can the 4 smaller squares be painted

Answer 1

The total number of ways the 4 smaller squares can be painted is 116,280.

Step-by-step explanation:

To find the number of ways the 4 smaller squares can be painted, we can start by considering the first square. Since it has 20 available colors to choose from, there are 20 possible choices for the first square. For the second square, since it is adjacent to the first square, we have 20-1=19 choices for its color. Similarly, for the third square, we have 19-1=18 choices, and for the fourth square, we have 18-1=17 choices. Therefore, the total number of ways the 4 smaller squares can be painted is 20 x 19 x 18 x 17 = 116,280 ways.

Answer 2

If we just take 4 of the paints there are 4! = 24 ways of painting the 4 squares.

We need to multiply this by 20C4 - the number of combinations of 4 from 20.

20C4 = (20*19*18*17) / (4*3*2*1) = 4845

so the answer is 24 * 4845 = 116,280