# Building a Burger

Pro Problems > Math > Statisics and Probabilities > Counting Principles## 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?

## Solution

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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!]

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 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?

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In how many ways can you arrange the digits?

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

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

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

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

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How many different ways could I have Thanksgiving Dinner?

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[No calculators, please]