There are 13 favors at the soda machine, and anyone can choose 6. A sign says we are only
allowed to use one of each favor. You decide to grab a cup and fill it up with your favorite favors
at random (it’s all going to the same place, right?). How many possible soda mixtures can you
make? Make sure to show your work and provide a numerical answer (i.e., don’t leave it in factorial
form).
To find the number of possible soda mixtures, we need to use the concept of combinations. A combination is a selection of items from a larger set without considering their order. In this case, we need to calculate the number of combinations of choosing 6 favors out of the 13 available.
The formula for calculating combinations is given by:
C(n, r) = n! / (r! * (n - r)!)
Where n is the total number of items available and r is the number of items to be selected.
In our case, we have 13 favors to choose from, and we want to select 6 favors. So, we can calculate the number of possible soda mixtures using the combination formula as:
C(13, 6) = 13! / (6! * (13 - 6)!)
Simplifying this expression:
C(13, 6) = 13! / (6! * 7!)
Now, we can calculate the value of 13! (factorial) and 6! (factorial) to get our numerical answer.
13! = 13 * 12 * 11 * 10 * 9 * 8 = 95,040
6! = 6 * 5 * 4 * 3 * 2 * 1 = 720
Substituting these values back into the combination formula:
C(13, 6) = 95,040 / (720 * 5040) = 95,040 / 3,628,800 = 0.0262 (approximately)
Therefore, there are approximately 0.0262 (or 26.2%) possible soda mixtures when choosing 6 favors out of the 13 available.