There are 13 flavors at the soda machine, and anyone can choose 6. A sign says we are only allowed to use one of each flavor. You decide to grab a cup and fill it up with your favorite flavors at random (it’s all going to the same place, right?). How many possible soda mixtures can you make?

Can someone show me step by step?

Is the answer 1716?

13C6 = 1716

Okay I just need help in putting it into exponential form. I forgot how to do that.

thank you.

Sure! Let's break down the problem step by step to find the number of possible soda mixtures.

Step 1: Determine the total number of flavors to choose from: in this case, there are 13 flavors available at the soda machine.

Step 2: Determine the number of flavors you can choose: you are allowed to choose 6 flavors.

Step 3: Calculate the number of possible soda mixtures.

To solve this, we can use the concept of combinations. The number of combinations can be found using the formula:

nCr = n! / (r!(n-r)!)

Where n is the total number of flavors and r is the number of flavors you can choose.

In this case, n = 13 (total flavors) and r = 6 (flavors to choose).

Plugging these values into the formula:

13C6 = 13! / (6!(13-6)!)
= 13! / (6! * 7!)

Now, let's simplify this expression:

13! = 13 * 12 * 11 * 10 * 9 * 8 * 7!
6! = 6 * 5 * 4 * 3 * 2 * 1
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1

Substituting these values into the expression:

13C6 = (13 * 12 * 11 * 10 * 9 * 8 * 7!) / ((6 * 5 * 4 * 3 * 2 * 1) * (7 * 6 * 5 * 4 * 3 * 2 * 1))

Canceling out common factors:

13C6 = (13 * 12 * 11 * 10 * 9 * 8) / (6 * 5 * 4 * 3 * 2 * 1)

Now, let's calculate the value:

13C6 = 1716

Therefore, you can make 1716 possible soda mixtures by choosing 6 flavors out of 13 at random.

I hope this breakdown helps you understand how to calculate the number of possible soda mixtures!