# Arby's and Combinatorics

1. Jul 4, 2007

Arby's has this deal out now: 5 for $5.95. You can pick from 8 choices to fill 5 spots. The menu says "over 790 possible combinations" , so one can assume that the number of choices is between 790 and 800. What is the equation to figure out how many choices you have and what is the exact number of possible combinations? 2. Jul 4, 2007 ### Werg22 There are well over 800 possibilities. 3. Jul 4, 2007 ### d_leet How much is well over 800? And how do you figure that out? 4. Jul 4, 2007 ### Werg22 Anyway to question is posed the number of possibilities is over 800; if one selection is independent of the others, there are 8^5 possibilities and if it is, there are 8*7*6*5*4 possibilities. 5. Jul 4, 2007 ### Integral Staff Emeritus Wiki has the answer. In this case, order does not matter and and you can order the same thing more then once. (2 Arby sandwichs and 3 frys) edit: I get 792 6. Jul 4, 2007 ### cristo Staff Emeritus The answer is not "way more than 790," but is actually 792. Check the "combination with repetition" section of this page of wikipedia. 7. Jul 4, 2007 ### d_leet All selections are independant, but 8^5 doesn't take into account any duplications of choices this is what I thought at first as well and realized it made no sense. 8. Jul 4, 2007 ### Werg22 Oops, sorry, I just realized. 9. Jul 4, 2007 ### sk381 Another related question.. Say I have 3 items A,B,C and I wish to select 3 at a time (order does not matter and an object can be chosen more than once)... using the forumla for Combination with repetition, I should have (3+3-1)!/3!*2!=10 combinations If I list all the possible combinations, I have: AAA BBB CCC AAB BBC AAC BCC ABB ACC But this is only 9.....what combination am I missing? 10. Jul 4, 2007 ### sk381 sorry.. didn't look at the most obvious choice.. ABC!!! 11. Jul 20, 2007 ### matticus lol wow i was just about to post a topic about this and i searched google first to see if anyone had. i was curious as to why they said over 790 choices, as that would seem to imply it's between 790 and 800 (or else they would have said over 800 choices). there are 8 choices, and 5 slots to put them in. that should be 8^5 if i remember from discrete math. 8^5 is a lot larger than 790 though. 12. Jul 20, 2007 ### cristo Staff Emeritus Read post #5 or #6. 13. Jul 22, 2007 ### cepheid Staff Emeritus Er, yeah, because the order of the elements doesn't matter. You're just choosing, not permuting. If you order two medium drinks, two curly fries and a roast beef sandwich, that's the same as ordering a roast beef sandwich, an order of curly fries, a medium drink, another order of curly fries, and a medium drink. You've put in the same food order, you just haven't requested the items in the same sequence. The two examples are not distinct combinations of five items. What Cristo said, if you want to see the mathematics. By the way, here in Canada, you still pay$6, but you only get to choose four items! I don't understand why that is. The menu says 330 possible combinations, which leads me to believe that there are still 8 items to choose from (if you use the formula provided in the Wikipedia link, it works out exactly).

That leads me to my next question. Does anyone know how to "derive" or at least intuitively explain this formula for combination with repetition? Because, all three of the other combinatorial formulas make perfect sense to me (and if you don't believe me, I'd be happy to explain why I think they make sense intuitively). But combination with repetitition...looking at the formula, I can't seem to figure out how one might arrive at it.