If a registration plate for a car consists of three letters

If a registration plate for a car consists of three letters (A-Z) and three numbers (0-9), in any arrangement, what is the probability that a registration plate will contain no duplicate letters or numbers?

Solution

There are 26^3 = 17576 ways to choose 3 letters.

There are 10^3 = 1000 ways to choose 3 digits.

There are 6!/(3!3!) = 20 ways to permut the letters and numbers.

Thus, a total of 17576*1000*20 = 351520000 total plates.

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If no letters repeat, there are 26P3 = 15600 ways to do that.

There are 10P3 = 720 ways so that no digit repeats.

There are 6!/(3!3!) = 20 ways to permut the letters and numbers.

Thus, a total of 15600*720*20 = 224640000 ways so that no letters/numbers duplicate.

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Hence,

P(all unique) = 224640000/351520000

= 0.639053254 [answer]