Choosing three pizza toppings from seven

Question

A pizzeria offers 7 toppings, and you will choose exactly 3 different ones. Putting mushroom then olive is the same pizza as olive then mushroom, so order does not matter. How many different topping combinations are possible?

Accepted Answer

Since the toppings form an unordered set of 3 chosen from 7, this is \binom{7}{3}: \binom{7}{3} = \frac{7 	imes 6 	imes 5}{3 	imes 2 	imes 1} = \frac{210}{6} = 35. There are 7 	imes 6 	imes 5 = 210 ordered ways to pick three toppings, but each set of 3 shows up in 3! = 6 orders, so we divide to get 35. Takeaway. Toppings are chosen as a set, so it is a combination: \binom{7}{3} = 35.

Choosing three pizza toppings from seven

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