Problem 65 - Algebra and Trigonometry

Question

Using 15 flavors of ice cream, how many cones with three different flavors can you create if it is important to you which flavor goes on the top, middle, and bottom?

Step-by-Step Solution

Answer

The number of different ice cream cones that can be created is 2730.

Step 1: Understand the Problem

We are given 15 unique flavors of ice cream, and we want to find the permutations of these flavors on an ice cream cone with three different positions (top, middle, bottom). A permutation is an arrangement of objects where the order is important.

Step 2: Use Permutation Formula

The formula for permutation is \(nPn = n! / (n - r)!\) where \(n\) is the total number of options, \(r\) is the number of selections made, and \(!\) denotes a factorial, meaning the product of all positive integers up to that number. Here, \(n = 15\) and \(r = 3\), as we are selecting 3 flavors out of 15 to place on the cone.

Step 3: Evaluate the Expression

Substitute the values into the formula. Evaluate \(15P3 = 15! / (15 - 3)!\). This can be further simplified by cancelling out terms, leaving us with \(15 × 14 × 13\). Find the product of these numbers.