# How many license plates can be made consisting of 2 letters followed by 4 digits?

The key observation that we need to deal with the ordering constraints is that there are $\binom{n}{k}$ ways to choose $k$ distinct elements out of $n$, and that for each of those choices there is only one way to arrange the $k$ elements so that they are ordered. Hence $\binom{26}{2}$ counts the ways to pick two letters in alphabetic order, and $\binom{10}{4}$ counts the ways to pick four digits in descending order.

If the first digit is allowed to be $0$, $\binom{10}{4}$ is also the number of ways to pick four digits in ascending order. This assumption is likely to hold for license plates; otherwise, $10$ must be replaced by $9$.

The solution that you already know tells you how to deal with distinct, but not ordered elements. If letters and digits may come in any order, for every choice of two distinct letters and four distinct digits there are $6!$ ways to arrange them. When the letters come before the digits, we reason similarly, but we separately count the ways to fill the letter positions and the ways to fill the digit positions.

For example, if the letters only need to be distinct, while the digits must be in decreasing order, we get $\binom{26}{2} \cdot 2! \cdot \binom{10}{4}$. Do you see how to continue?

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## 2 Answers By Expert Tutors

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letter letter digit digit

26  * 26  * 10 * 10=67,600 license plates

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The answer would be = 2^2 = 4

since repetition is allowed the number of licence plates you could make is 4.

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Solution:

Given, license plates consist of 3 letters followed by 2 digits.

Let the numbers on license plates be N

Let the letters on license plates be L

So, the license plate consisting of 3 letters and 2 digits will be LLLNN.

Letters can be anything from A to Z.

There are 26 letter combinations for the first letter. Again second and third letters can be anything from the 26 letters.

So, combination for letters = 26 × 26 × 26

= 17576

Numbers can be anything from 0 to 9.

There are 10 combinations for each place.

So, the combination for numbers = 10 × 10 = 100

Now, the combination for letters and numbers = 17576 × 100 = 1757600.

## How many license plates can be made consisting of 3 letters followed by 2 digits?

Summary:

1757600 license plates can be made consisting of 3 letters followed by 2 digits.

 Click here to see ALL problems on Miscellaneous Word Problems Question 1134623: A license plate has 4 letters, followed by 2 numbers. If repetition is not allowed, how many different license plates can be made? If repetition is allowed, how many different license plates can be made? Found 2 solutions by Theo, ikleyn:Answer by Theo(12345)   (Show Source): You can put this solution on YOUR website! the license plate has 4 letters followed by 2 numbers. there are 26 possible letters for each letter position. there are 10 possible numbers for each number position. the total number of license plates will be equal to 26^4 * 10^2 = 45,697,600. if you assume small letters and big letters can be used, then the number of possible license places will be 52^4 * 10^2 = 731,161,600. nomall, only capital letters are used, so your answer should be 45,697,600 license plates can be issued. Answer by ikleyn(46540)   (Show Source): You can put this solution on YOUR website! .If repetition is not allowed, then the number of different licence plates is 26*25*24*23*10*9. If repetition is allowed, then the number of different licence plates is . 

### How many license plates of 4 symbols can be made using 2 letters and 2 digits?

How many license plates of 4 symbols can be made using 2 letters and 2 digits? The answer is 405,600.

### How many license plates can be made using 2 letters followed by 3 digits?

So for a license plate which has 2 letters and 3 digits, there are: 26×26×10×10×10=676,000 possibilities.

### How many license plates can be made with 2 letters and 2 digits?

2 Answers By Expert Tutors since repetition is allowed the number of licence plates you could make is 4.