Question

In how many ways can a person order one ice cream cone with 3 different

flavors of ice cream if there are 14 flavors to choose from and it matters to
the person how the 3 flavors are stacked on the one cone? That is, which
flavor is on the top, middle, and bottom.

Answer 1

2184

Explanation:

You want to know the possible number of ways 3 flavors can be chosen from 14 when order matters.

Permutations

The number of permutations of 14 flavors taken 3 at a time is ...

14 × 13 × 12 = 2184

There are 14 ways to choose the bottom flavor, then 13 ways to choose a different middle flavor, and finally 12 ways to choose a different top flavor. Each flavor can go with any combination of the others, so the total number is the product of 14·13·12, or 2184.

The formula for the number of n things taken k at a time is ...

P(n, k) + n!/(n-k)!

For n=14, k=3, this is the product shown above.