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?

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?

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?

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?

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.
• You can have it with cheddar cheese, monterey jack cheese, or no cheese.
• 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?

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