Combinations of Toppings when Ordering a Pizza
Date: 05/19/2005 at 12:08:04 From: Tom Subject: Combinations/Permutations How many combinations of pizza can be made with 6 different toppings? Assuming that double toppings are not permitted, can you explain why the answer is 2^6? Thanks. I get the same answer using c(6,0) + c(6,1) + c(6,2) + ... + c(6,6), but I can't understand why 2^6 works other than that both = 64.
Date: 05/27/2005 at 09:49:26 From: Doctor Wilko Subject: Re: Combinations/Permutations Hi Tom, Thanks for writing to Dr. Math! I was confused by this answer when I first saw it in a statistics class too. But the reasoning is of a binary nature. You can either add the topping or not. Your solution of C(6,0) + ... + C(6,6) is probably more intuitive at first, but it turns out both answers are correct. I can ask if you want each of these toppings on your pizza and you can give me one of two answers: Cheese: Yes or No. 2 answers Peppers: Yes or No. 2 answers Olives: Yes or No. 2 answers Sausage: Yes or No. 2 answers Anchovies: Yes or No. 2 answers Onions: Yes or No. 2 answers Therefore, the answer is: 2 * 2 * 2 * 2 * 2 * 2 = 2^6 = 64 different pizzas You've made a neat connection with the combinations that I also discovered on my own and was pretty excited about when I first saw it. 2^n = C(n,0) + ... + C(n,n) This connection can be more obvious if you see how it fits into Pascal's Triangle and Combinations. I'll provide a link below for you to look at. Knowing this connection just gives you another tool that you can use to solve problems like this. You'll find with counting problems that there are usually multiple ways to get to the answer. Feel free to visit our archives for more insight on this topic: Permutations and Combinations http://mathforum.org/dr.math/faq/faq.comb.perm.html Pascal's Triangle http://mathforum.org/dr.math/faq/faq.pascal.triangle.html Does this help? Please write back if you have questions. - Doctor Wilko, The Math Forum http://mathforum.org/dr.math/
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