# Buttons and Levers

Pro Problems > Math > Statisics and Probabilities > Counting Principles## Buttons and Levers

There are three buttons on a wall, each of which can be pressed or unpressed. There are four levers on the wall, and each of them can be either flipped up or down, or in a central position. How many possible arrangements of levers and buttons are there?

## Solution

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### How Many Arrangements?

The number 5435 is made up of the digits 3, 4, and 5, with the 5 used two times. The numbers 3455 and 5534 are two examples of other ways to arrange these digits.

In how many ways can you arrange the digits?

### 161 - Digits and Letters

In a basket I have slips of paper with several different digits on them. In another basket I have slips of paper with several different letters on them. If I reach into the first basket and pull out a number, and then reach into the second basket and pull out a letter, there are 161 possible results. How many letters of the alphabet are *not *in the second basket?

### Building a Burger

I can have either mustard, ketchup, or relish for my burger. I can choose either swiss, Monterey, or cheddar cheese. I can choose either pickles, tomatoes, or lettuce. I can choose wheat bread or white (but I must have bread). I dislike the combination of mustard and swiss cheese. I like all other combinations, including combinations which include no cheese, condiments or veggies. How many combinations are there which I will like?

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

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On an island with seven towns, one town borders on five of the other towns, three towns border on four other towns, two towns border on three other towns, and one town only borders one other town.

Every day postmen from each town walk to each of their borders and exchange a sack of mail with a postman from the neighboring town. How many such exchanges happen each day?

### 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?

### Twelve Days of Christmas

We all know the song "The Twelve Days of Christmas," and how the gift giving works:

Day One: 1 partridge in a pear tree

Day Two: 2 turtle doves *and *1 partridge in a pear tree

Day Three: 3 calling birds, 2 turtle doves, and 1 partridge in a pear tree

Thus, after three days, the singer has 1 + 3 + 6 = 10 gifts. [Note that some people interpret the song as 1 gift the first day, 2 the second day, three the third day, etc., but that is not what the song *says*, so we will go with the literal interpretation of the song!]

After Mrs. Claus heard the elves singing this song one Christmas, she decided that the next year she would take this song to the extreme, and extend it to 20 days of Christmas, giving gifts (in the same pattern as the song) for 20 days.

Santa Claus caught wind of what she was doing and, since he is considered to be the epitome of the spirit of giving at Christmas time, decided he couldn't be out-given, and did the 25 days of Christmas instead.

By how many gifts did Santa "outgive" Mrs. Claus?

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Burger Heaven is the new restaurant in town, and they offer to make your burger however you want it, with the following possibilities:

- You can have your burger rare, medium rare, medium well, or well done.
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- You can have it with or without mustard.
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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?

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