12 different flavors of ice cream and 2 different toppings. How many different kinds of sundaes could be made if each sundae contained 2 scoops of ice cream and one topping?
C(12,2)*2
assuming the scoops must be different flavors. If duplicate scoops are allowed, then it's just
12^2 * 2
To determine the number of different kinds of sundaes that can be made, we need to calculate the combinations of ice cream flavors and toppings.
First, let's consider the combinations of ice cream flavors. We have 12 different flavors, and we need to choose 2 for each sundae. This can be calculated using the combination formula, denoted as C(n, r), where n is the total number of options and r is the number of options to choose:
C(12, 2) = 12! / (2! × (12 - 2)!)
C(12, 2) = 12! / (2! × 10!)
C(12, 2) = (12 × 11) / (2 × 1)
C(12, 2) = 66
So, there are 66 different combinations of ice cream flavors for each sundae.
Next, let's consider the number of options for toppings. We have 2 different toppings, and we need to choose 1 for each sundae. Therefore, there are 2 options for each sundae.
To calculate the total number of different kinds of sundaes, we multiply the number of flavor combinations by the number of topping options:
66 × 2 = 132
Hence, there can be 132 different kinds of sundaes made using the given criteria.