# 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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### 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?

### Tic Tac Toe Grid

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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?

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In how many ways can you arrange the digits?

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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!]

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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:

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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!"

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[No calculators, please]

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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?

### Making License Plates

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