A pack of Jellybeans contains 7 different flavors: Apricot, Banana, Coconut, Date, Eggplant, Fig, and Grape. You can Eat each flavor individually or come up with some crazy combinations. How many flavor combinations are possible with these 7 flavors alone or mixed? You can use only use one flavor per combination. Example, no ABBC, but ABDEF is good.
To find out how many flavor combinations are possible with these 7 flavors, we need to consider both individual flavors and mixed combinations.
The number of individual flavor combinations is simply the number of flavors, which is 7.
For mixed combinations, you can choose any number of flavors from 1 to 7 at a time. Let's break it down:
- When choosing 1 flavor at a time, you have 7 choices.
- When choosing 2 flavors at a time, you have 7 choices for the first flavor and 6 choices for the second flavor (since you cannot repeat flavors).
- When choosing 3 flavors at a time, you have 7 choices for the first flavor, 6 choices for the second flavor, and 5 choices for the third flavor.
- This pattern continues until you choose all 7 flavors at a time, where you would have 7 choices for each flavor.
To find the total number of mixed combinations, you sum up all the possible combinations for each selection size:
7C1 + 7C2 + 7C3 + 7C4 + 7C5 + 7C6 + 7C7
Using the formula for combinations (nCr), where n is the total number of items and r is the number of items chosen at a time:
7C1 = 7
7C2 = 7! / (2! * (7-2)!) = 21
7C3 = 7! / (3! * (7-3)!) = 35
7C4 = 7! / (4! * (7-4)!) = 35
7C5 = 7! / (5! * (7-5)!) = 21
7C6 = 7! / (6! * (7-6)!) = 7
7C7 = 7! / (7! * (7-7)!) = 1
Adding up the combinations:
7 + 21 + 35 + 35 + 21 + 7 + 1 = 127
So, there are 127 possible flavor combinations with these 7 flavors alone or mixed.