License Plate - Probability Conditional

Question

Hello,

What is the probability of getting a license plate that has a repeated letter or digit if you live in a state in which license plates have two letters followed by four numerals? (Round your answer to one decimal place.)

The answer is 51.5%. Can someone show me the solution to this?

Answer 1

How many ways can we choose the elements ?

As there are 26 letters we can choose the letters in 26 x 26 ways

and as there are 10 digits 0-9 we can choose the digits in 10 x 10 x 10 x 10 ways

so in total 26 x 26 x 10 x 10 x 10 x 10 ways = 676 0000 ways

Suppose letters and digits are not repeated

Then we can choose the letters in 26 x 25 ways

and we can choose the digits in 10 x 9 x 8 x 7 ways

so total = 26 x 25 x 10 x 9 x 8 x 7 ways = 3276000 ways

so ways in which a letter or digit is repeated = 676 0000 - 3276 000 = 3484 000

so probability = 3484000 / 6760000 = 0.5154 = 51.5 %