# 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

In order to make it feasible for teachers to use these problems in their classwork, no solutions are publicly visible, so students cannot simply look up the answers. If you would like to view the solutions to these problems, you must have a Virtual Classroom subscription.## Similar Problems

### Tic Tac Toe Grid

Begin with a 3 x 3 grid, like the one used to play Tic Tac Toe. You have a blue square, a green square, and a red square, that you can place within the grid, according to the following rules:

- 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.
- No square can be placed on top of another square.

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?

### 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?

### 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?

### 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?

### STOPS!

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

### Andy and Roger

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?

### 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?

### Borders

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?

### Guitar Pickin'

Owen is picking out random tunes on his guitar. He can play either half notes, quarter notes, or eighth notes. Each pluck can be one of the tones of the scale (C, C#, D, D#, E, F, F#, G, G#, A, A#, B). If he wanted to play every possible three-note tune, and he could do one tune every 20 seconds, how many days would it take him to play all of them?

[No calculators, please]