# How many plates can be made?

How many vehicle license plates can be made if the licenses contains 2 letters of the English alphabet followed by a three digit number.

1. If repetitions are allowed.
2. If repetitions are not allowed.
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Prem, it is important to know why you answer is correct. Please ask questions. I have given a simple explanation, Dennis has used the proper notation that you should learn use once you have grasped the concept. – futurebird Apr 22 '11 at 13:15
i have a suspicion about the 1st question whether it will be 26*26*999 Or 26*26*1000.........I am not getting why the text book is used 999 – prem shekhar Apr 22 '11 at 13:21

#1

The licence plate is described in-order so you can count the possibilities in order. Here is your plate:

__ __ __ __ __

What can go in the first spot? (ABCDEFGHIJKLMNOPQRSTUVWXYZ) So, there are 26 possibilities. What about the next spot? Again (ABCDEFGHIJKLMNOPQRSTUVWXYZ) .. so 26 again:

26 * 26 * __ __ __ = ?

now use the same idea for the spaces with numbers.

#2

Next, how do we count plates when you cannot repeat? If we have picked one of the 26 letters of the alphabet for the first element... then how many choices do we have for the next?

26 * 25 * __ __ __ = ?

Now, for the numbers. We have 10 possibilities as in the previous problem. (0123456789) But after we have used one number for the next slot there will only be 9 possibilities.

26 * 25 * 10 * 9 __ = ?

In fact, what we are doing is using a formula known as the "choose formula" or the binomial coefficient.

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Ya ya for the 1st question answer should be 26*26*10*10*10 but given answer in text book is 26*26*999 – prem shekhar Apr 22 '11 at 13:17
you are right at second question – prem shekhar Apr 22 '11 at 13:17
That sounds like an error. Like they forgot that 0 is a number... – futurebird Apr 22 '11 at 13:22
i think they are considring 000 not a number – prem shekhar Apr 22 '11 at 13:23
Although to be fair, O and Q can be hard to distinguish from 0, and I is hard to distinguish from 1, so our expected number of ways to letter a license plate would be lower. A good exercise for the poster is to find how many ways to letter a license plate with these restrictions in place. – Joshua Shane Liberman Apr 22 '11 at 13:59

1) If repetitions are allowed, then you have $26^2$ options to pick 2 letters, and $10^3$ to pick three digits. Total of $26^2\cdot 10^3$

2) If repititions are not allowed, then $\binom{26}{2}$ options to pick 2 letters, and $\binom{10}{3}$ to pick three digits. Total of $2\cdot\binom{26}{2}\times3!\cdot \binom{10}{3}$

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1)if repetition then 26^2 *999 will be the answer...that is what i am geeting from textbook....do you think this answer is right? – prem shekhar Apr 22 '11 at 13:13
2)If No repetition then 26P2*10P3=468000 – prem shekhar Apr 22 '11 at 13:14