Permutation and combination related

Question

A license plate consist of 3 digits followed by 3 letters. how many different number of license plates can be made if a digit or a number can be used twice?

Answer 1

From the wording of the problem, I assume this means that no character can be "three-peated" (so, for example, 222ABC or 345 XXX) are illegal.

 

The answer is the number of legal digit groups times the number of legal letter groups. 

 

There are 10 x 10 x 10 digit groups possible, of which 10 (111, 222, 333...) are illegal. This means that there are 

10 x 10 x 10 - 10 = 1,000 - 10 = 990 legal digit groups.

 

There are 26 x 26 x 26 letter groups possible, of which 26 (AAA, BBB, CCC...) are illegal.

This means that there are
26 x 26 x 26 - 26 = 17,576 - 26 = 17,550 legal letter groups. The number of legal license plate numbers is 990 x 17.550 = 17,374,500.