# Arby's 5 for \$5.95 (combinations)

Discussion in 'Physics & Math' started by cato, Aug 11, 2005.

1. ### catoless hate, more scienceRegistered Senior Member

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hey everyone, I just saw an Arby's commercial and it said "do the math" which I planned to do (I know I am playing into their strategy), but I am having trouble remembering how to figure out the number of combinations.

I am a bit fuzzy, but its either 5 of 5 different kinds of sandwiches, or 5 of 2 different kinds of sandwiches. could someone remind me how to calculate the number of possible food combinations with these two deals?

my question is admittedly a bit confusing, if you need explanation, feel free to ask.

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3. ### DinosaurRational SkepticValued Senior Member

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Combinations(n, r) = n! / (n - r)! * r!

For 5 distinct items 2 at a time = 5*4*3*2*1 / (3*2*1)*(2*1)
= 5*4 / 2
= 10

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5. ### DarkEyedBeautyPirate.Registered Senior Member

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Are you kidding? It's just a ploy to get you to eat more of their disgusting addictive food. I'm sure if they're pricing their sandwiches so cheaply, they are obviously a cheaply made product.

I would never waste my money.

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7. ### UnderWhelmedRegistered Senior Member

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I'll stick to McDonalds!

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8. ### AerRegistered Senior Member

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Yes, McDonalds >> Arbys

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9. ### AerRegistered Senior Member

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I appears to be 5 of 5 different kinds of sandwiches assuming there are at least 4 kinds of "Melts"

(well now I feel like I was duped by this ploy for seeking at their advertisement)

10. ### catoless hate, more scienceRegistered Senior Member

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gerr, I don't care, I never eat there anyway, and I wont now, I just want the math. by the way, you can use a factorial, I understand them. so what are the combinations for 5 of 5 different sandwiches?

11. ### AerRegistered Senior Member

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I can't answer your question as I have no idea how many distict sandwiches there are.

12. ### RubiksMasterReal eyes realize real liesRegistered Senior Member

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5 of 5 different sandwiches? I may be misunderstanding, but isnt it just 5*5 = 25?

13. ### DilbertRegistered Senior Member

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or simply 5. as in 5 out of 5. or 5 choices with 5 different types of sandwiches.

14. ### AerRegistered Senior Member

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But you could get 5 of sandwich A or you could get 4 of sandwich A and 1 of sandwich B or you could get 3 of sandwich A and 2 of sandwich B or you could get 1 of sandwich A and 1 of sandwich B and 1 of sandwich C etc.

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17. ### catoless hate, more scienceRegistered Senior Member

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it is 5 different kinds of sandwiches. and you get five of your choice of the 5 sandwiches. so you get 5 sandwiches, and you can pick through 5 different kinds to fill out that order of 5. does that make sense? its hard to explain. so it would be 5^5=(3125)

thanks UnderWhelmed.

Last edited: Aug 11, 2005
18. ### Cottontop3000Death BeckonedRegistered Senior Member

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Shut UP!!! YOU'RE MAKING ME HUNGRY FOR THAT SHITE!!

19. ### shmoeRegistred UserRegistered Senior Member

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There are 5 kinds of sandwiches to pick from, and you have 5 choices? The answer is not 5^5 then. 5^5 is making the order you recieve the sandwiches distinct (or they are going to be eaten by 5 different "victims"), which it didn't seem like you were after?

You can repair this with inclusion/exclusion or go a different and much easier way. You can think of it as an arrangement of 5 s's and 4 vertical lines |. Each "s" before the first "|" means a sandwich 1, each "s" between the next 2 "|" means a sandwich 2, and so on. eg.

s|s|s|s|s means one of each gross sandwich
sssss|||| means 5 of gross sandwich #1
ss|ss|||s means 2 gross sandwich #1, 2 icky sandwich #2 and 1 disgusting #5

In this way, the number of distinct strings of 4 "|" and 5 "s" is equal to the number of distinct sandwich orders. Counting these strings is easy, you can think of it as choosing 4 of the 9 spots for the "|", so it's "9 choose 4":

9!/(4!*(9-4)!)=126

(thinking of it as choosing 5 of the 9 spots for the "s"'s will give the same answer)

20. ### DinosaurRational SkepticValued Senior Member

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I really do not see any definition of the problem in this thread. A possible definition is the following.
• There are 5 different kinds of sandwiches, and for some price you can order five of them.

• For example one of each.

• Another example: Two of this type & and three of that type
The following analysis seems correct for the above definition of the problem.
• 1 1 1 1 1 (one of each). there is only one combination.

• 2 1 1 1 0 (Two of one kind and one of three other kinds). There are 5 possibilities for the type ordered twice, and four choices for the type not ordered at all. This looks like 20 possibilities.

• 3 1 1 0 0 This looks like 60 possibilities. 5 choices for the type ordered three times, and 12 possibilities for the types not ordered at all (4 times 3).

• 4 1 0 0 0 This looks the same as 2 0 1 1 1 or 20 possibilities.

• 5 0 0 0 0 This looks like 5 possibilities.

• 2 2 1 0 0 This looks the same as 1 1 3 0 0 or 60 possibilities.

• 3 2 0 0 0 This looks the same as 4 1 0 0 0 and 2 0 1 1 1 or 20 possibilities.
The above adds up to 186 possibilities, and I do not think I omitted a possible order.

Let me know if I overlooked somwthing or made an arithmetic error.

21. ### shmoeRegistred UserRegistered Senior Member

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This count is off, there are 5 choices for whatever is ordered three times, then "4 choose 2"=4!/(2!*2!)=6 ways to select the 2 not ordered, for a total of 30

as above, this should be 30, net result is you're 60 too high, so 186-60=126. You've interpreted it the same as I did in my last post, this seems to be what cato was looking for (though we could both be wrong of course!).

22. ### AerRegistered Senior Member

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I prefer the first answer, 3125, because I can order them in which ever order I want. This means when confirming the order, I can switch the order up. Kinda like ordering a strawberry milkshake, minus the strawberry.

23. ### shmoeRegistred UserRegistered Senior Member

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Sure, but what the order that you place your order is irrelevant to what sandwiches you walk away with, and the type of gastronomical distress you'll be in later. We need clarification from cato on what he's actually after.