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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 

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 

  Pascal's Triangle 

Does this help?  Please write back if you have questions.

- Doctor Wilko, The Math Forum 
Associated Topics:
High School Permutations and Combinations

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