Ben decides to get an ice cream cone. He is going to choose two toppings out of an available six. Ben also has to choose between a plane cone or a waffle cone. If he chooses vanilla ice cream how many different combinations of toppings and cones does he have to choose from?
2 * 6C2
To find the number of different combinations of toppings and cones that Ben can choose, we need to multiply the number of options for each choice together.
First, we need to determine the number of options for choosing two toppings out of six available toppings. This requires using the combination formula. The formula for combination is n C r = n! / (r! * (n - r)!), where n is the total number of items to choose from, and r is the number of items to be chosen at a time.
In this case, n = 6 (available toppings) and r = 2 (toppings to be chosen). Plugging these values into the combination formula, we get:
6 C 2 = 6! / (2! * (6 - 2)!) = 6! / (2! * 4!) = (6 * 5 * 4!) / (2! * 4!) = (6 * 5) / (2 * 1) = 15
So, there are 15 different ways Ben can choose two toppings from the available six toppings.
Next, Ben has to choose between a plain cone or a waffle cone. Since he can only choose one type of cone, there are 2 options.
Finally, to find the total number of combinations, we multiply the number of topping combinations by the number of cone choices:
15 (topping combinations) * 2 (cone choices) = 30
Therefore, Ben has 30 different combinations of toppings and cones to choose from if he chooses vanilla ice cream.