How many different codes are possible?

Question

Assume that X={A,B,G}X={A,B,G} and Y={2,7,3,1,4,5}Y={2,7,3,1,4,5}. A code consists of 2 different symbols selected from XX followed by 3 not necessarily different symbols from YY.

How many different codes are possible?

Answer 1

There are three ways of selecting the first letter. You can select A, B, or G. Once you select one letter, you can't select the same letter, so therefore there will be only two ways of selecting the second letter. For example, if you selected the first letter as B, you could only select A or G to be the second letter. Therefore, there are 3 x 2 = 6 ways of selecting the first two letters. Now, we'll use the same logic to select the numbers. There are 6 ways of selecting the first number. The numbers don't have to be different, so there are also 6 ways of selecting the second number and six ways of selecting the third number. Thus the total possibilities are 3 x 2 x 6 x 6 x 6 = 1296 ways