Question

A license plate contains 3 letters in all caps followed by four numbers from (0,1,...9).

A. Find the probability that the four numbers are all distinct on a license plate.

B. Find the probability that there are three distinct letters and exactly two 5's on a license plate

Answer 1

Answer a)

There are 3 letters in licence plate, each of which can be selected in 26 ways. This gives:

26*26*26

There are four numbers in licence place. But since number are all distinct. The first first number can be any one of the 10 digits. The second number can be any one of the remaining 9 digits. The third number can be any one of the remaining 8 digits. And fourth number can be any one of the remaining 7 digits. This gives:

10*9*8*7

Total no. of ways in which license plate contains all distinct numbers = 26*26*26*10*9*8*7

Total no, of ways in which licence plate can be formulated = 26*26*26*10*10*10*10 (Repetition of no. also allowed)

Probability = (26*26*26*10*9*8*7)/(26*26*26*10*10*10*10)

Probability = 0.504

Answer b)

There are three letters in licence place. But since letters are all distinct. The first first letter can be any one of the 26 letters. The second letter can be any one of the remaining 25 letters. And the third letter can be any one of the remaining 24 letters. This gives:

26×25×24=15600

For numbers, it is given that there should be exactly two 5's. So, we select 2 letters out of 4 where two 5's can be placed. Remaining 2 places can be filled in 9 ways each. This is because we cannot use 5 in other places. This gives:

₄C₂*9*9 = 6*9*9 = 486

So, total number of ways in which licence place contains three distinct letters and exactly two 5's = 15600*486

Total no, of ways in which licence plate can be formulated = 26*26*26*10*10*10*10 (Repetition allowed)

Probability = (15600*486)/(26*26*26*10*10*10*10)

Probability = 0.0431