(a) A pizza parlor has a choice of 12 toppings for its pizzas. From these 12 toppings, how many different 7-topping pizzas are possible?(b) From a collection of 51 store customers, 3 are to be chosen to receive a special gift. How many groups of 3 customers are possible?
Question
(a) A pizza parlor has a choice of 12 toppings for its pizzas. From these 12 toppings, how many different 7-topping pizzas are possible?(b) From a collection of 51 store customers, 3 are to be chosen to receive a special gift. How many groups of 3 customers are possible?
Solution
(a) The number of ways to choose 7 toppings from 12 is given by the combination formula C(n, k) = n! / [k!(n-k)!], where n is the total number of options, k is the number of options to choose, and "!" denotes factorial.
In this case, n = 12 (the total number of toppings) and k = 7 (the number of toppings to choose for each pizza).
So, the number of different 7-topping pizzas is C(12, 7) = 12! / [7!(12-7)!] = 792.
(b) The number of ways to choose 3 customers from 51 is also given by the combination formula.
In this case, n = 51 (the total number of customers) and k = 3 (the number of customers to receive a special gift).
So, the number of groups of 3 customers is C(51, 3) = 51! / [3!(51-3)!] = 19600.
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