# Making License Plates

Pro Problems > Math > Statisics and Probabilities > Counting Principles## Making License Plates

In the State of Confusion, all car license plates are made with the following stipulations:

- The license plate will have six characters.
- The first character will be a digit between one and nine inclusive.
- The next two characters will be digits between zero and nine inclusive.
- The fourth character will be a vowel
- The fifth character will be a consonant
- The sixth character could be any letter.

How many different license plates are possible in the State of Confusion?

## Solution

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- You can have it with or without mustard.
- You can have it with or without ketchup.
- You can have it with iceberg lettuce, romaine lettuce, or no lettuce.
- You can have it with pickles or without pickles.

Joan and Gene each order a burger. Joan says to the waiter, "I hate cheese, and I don't like romaine lettuce, but other than that, I'll take it however you feel like preparing it. Surprise me!"

Gene says to the waiter, "I want my burger well done, and with mustard and ketchup. Other than that, I don't care how it's prepared - surprise me!"

The waiter brings the two burgers on a tray. What is the total number of possibilities for what could be on the tray?

### Thanksgiving Dinner

For Thanksgiving, I will have either have ham or turkey.

If I have turkey, I will have either potatoes or stuffing.

If I have ham, I will have either potatoes or rice.

If I have potatoes, I will have either green beans or peas.

If I have rice, I will have either peas or carrots.

If I have stuffing, I will have either carrots or green beans.

If I have green beans, I could have cranberry sauce with it.

If I don't have grean beans, I could have green bean casserole.

If I don't have peas, I can choose to have spinach.

For dessert I will have either ice cream or pie.

If I have ice cream, I could either have chocolate sauce or whipped topping on it, or put nothing on it.

If I have pie, I can choose mince, lemon meringue, cherry, or apple.

How many different ways could I have Thanksgiving Dinner?

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Andy picks three letters from his name and arrange them on the table. Roger picks two letters from his name and arrange them. If Roger puts his two letters after Andy’s three letters, how many possible arrangements of five letters are there?

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In how many ways can you arrange the digits?

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In how many different ways can the letters STOPS be arranged?

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- The green square must be touching the blue square on a side.
- The red square must be touching the green square on either a side or a vertex.
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For example, the following arrangement is valid, because the green is horizontally connected to the blue square, and the red square is diagonally connected to the green square.

In how many different ways could the three squares be placed on the grid?