You have heard the commercial for the restaurant that lets you pick any 3 appetizers, any of the 6 entree items and then a dessert. You have been told there are 72 different meal options. How many desserts are available?
I don't understand how to calculate this. I know how to calculate combinations, but I don't know how to work this specific problem. if you could give me a few steps to get started that would be greatly appreciated. thank you.
Let the number of desserts be x
the number of ways to choose a meal would be
(3)(6)(x)
but that is supposed to be 72
18x = 72
x = 72/18 = 4
To solve this problem, we can break it down into three parts: the number of options for appetizers, the number of options for entrees, and the number of options for desserts. Let's start by finding the number of options for appetizers. The commercial states that you can pick any 3 appetizers. This means that we need to find the number of combinations of 3 items from the available appetizer options. Since the order doesn't matter, we can use the combination formula.
To calculate the number of combinations, you can use the formula:
C(n, r) = n! / (r! * (n-r)!),
where n is the total number of items and r is the number of items to be selected.
In this case, if you have a total of x appetizer options, you can calculate the number of combinations of 3 appetizers using the formula:
C(x, 3) = x! / (3! * (x-3)!)
Next, let's determine the number of options for entrees. The commercial states that there are 6 entree items available. Since you can choose any one of these items, the number of options for entrees is simply 6.
Finally, let's find the number of options for desserts. The problem states that there are a total of 72 different meal options, which means that the combination of appetizers, entrees, and desserts must total 72. Since we have already determined the number of options for appetizers and entrees, we can subtract their combinations from 72 to find the number of options for desserts:
Number of options for desserts = 72 - (number of options for appetizers) * (number of options for entrees)
Substitute the values you calculated for the number of options for appetizers and entrees into the formula to find the number of options for desserts.
I hope this explanation helps you understand how to approach the problem and calculate the number of dessert options.