# Ways of ordering a hamburger

1. Oct 4, 2011

### Zee88

Hi. I thought this would be a great time to open with my first ever question on physics forums. =)

I´ve got stuck with a question regarding introductory Statistics.

The problem is:

Several years ago, Wendy´s Hamburgers advertised that there are 256 different ways to order your hamburger. You may choose to have, or omit, any combination of the following on your hamburger: mustard, ketchup, onion, pickle, tomato, relish, mayonnaise, and lettuce. Is the advertisement correct? Show how you arrive at your answer.

I actually dont know how to tackle the issue..

Can someone just give me a hint of how to proceed?

Thanks! =)

2. Oct 4, 2011

### daveb

Set the problem aside for a moment and look at the choices if there is only 1 item, say mustard. How many choices can you have? Now try 2 ingredients. Now try 3. Notice a pattern?

3. Oct 4, 2011

### Zee88

I´ve tried that. It didn´t help me =(

1 item = 1 combination or (1!)

2 items = 2 combinations or (2!)

3 items = 6 combinations or (3!)

4 items = 24 combinations or (4!)

5 items = 120 combinations or (5!)

By this standard 7 items should have 5040 different combinations? (7!) =/

4. Oct 4, 2011

### DaveC426913

(I remember decades ago, the same thing happened with the Rubik's Cube. It was advertized as having billions of combinations that it could be solve with. In fact, the real number is more than a billion times larger than that - 43 quintillion.)

But back to the question...

With 2 toppings there are actually 4 permutations:
KM
K-
-M
--

With 3 toppings there are actually 8 permutations:
KMR
KM-
K-R
-MR
K--
-M-
--R
---

Last edited: Oct 4, 2011
5. Oct 4, 2011

### Zee88

Thanks! =)

The book says the ad is correct. I´m guessing they missed something there.. =)

6. Oct 4, 2011

### DaveC426913

Well, 22 is 4, like I said.
Well, 23 is 8, like I said.
What's 28?

7. Oct 4, 2011

### Zee88

Oh okey! Now I get it! Thanks!

2 toppings = 2^(2) = 4 permutations

3 toppings = 2^(3) = 8 permutations

...

8 toppings = 2^(8) = 256 permutations

8. Oct 4, 2011

### Ray Vickson

That is the right idea, but these are not PERMUTATIONS; they are just "possibilities". Permutations refer to sorted arrangements, so would count the different ways of putting on the extras (eg., tomato first, onion second, mustard third, vs. onion first, mustard second, tomato third, etc.)

RGV

9. Oct 4, 2011

### DaveC426913

Mustard then ketchup is a distinct burger from ketchup then mustard? No, that's silly.

10. Oct 4, 2011

### SteamKing

Staff Emeritus
Math looks at two distinct kinds of ordering: permutations and combinations.

Permutation is selecting things from a larger group taking into account the order in which the items are selected. For example, putting ketchup on the burger first then the mustard is one permutation, and putting mustard on before the ketchup is another distinct permutation.

Combination is selecting things from a larger group without regard to the order of selection. A burger with ketchup and mustard is one distinct combination, and it doesn't matter which topping went on first.

11. Oct 4, 2011

### DaveC426913

Yes. Indeed, I was sloppy calling them permutations.

That being said, let's keep our eye on the ball. Ketchup then mustard being different from mustard then ketchup is a preposterous assumption.

12. Oct 5, 2011

### Ray Vickson

Yes, it is silly. And that's why the word "permutation" was wrong. Maybe it is not a big deal, but we are dealing here with a *student* who may be unfamiliar with these matters and be struggling with the concepts; that is why I thought it a good idea to be accurate.

RGV

13. Oct 5, 2011

### NewtonianAlch

The real question is why have ketchup and mustard on the burger at the same time? That's just disgusting!