A license plate contains 3 let
A license plate contains 3 letters in all caps followed by fournumbers from (0,1,…9).
A. Find the probability that the four numbers are all distincton a license plate.
B. Find the probability that there are three distinct lettersand exactly two 5’s on a license plate
Answer:
Answer a)
There are 3 letters in licence plate, each of which can beselected in 26 ways. This gives:
26*26*26
There are four numbers in licence place. But since number areall distinct. The first first number can be any one of the 10digits. The second number can be any one of the remaining 9 digits.The third number can be any one of the remaining 8 digits. Andfourth number can be any one of the remaining 7 digits. Thisgives:
10*9*8*7
Total no. of ways in which license plate contains all distinctnumbers = 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 areall distinct. The first first letter can be any one of the 26letters. The second letter can be any one of the remaining 25letters. And the third letter can be any one of the remaining 24letters. 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 wecannot use 5 in other places. This gives:
4C2*9*9 = 6*9*9 = 486
So, total number of ways in which licence place contains threedistinct 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
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