Problem 40 - Precalculus with Limits

Question

In how many ways can six people sit in a six-passenger car?

Step-by-Step Solution

Answer

There are 720 ways for six people to sit in a six-passenger car.

Step 1: Understanding the concept of permutations

A permutation is an arrangement of objects, without repetition, in a specific order. Since we are arranging people in the car and the order is important (i.e., sitting in the driver's seat is different from sitting in the passenger seat etc.), this is a permutation problem. In general, the number of permutations of n objects taken r at a time is given by \( nPr = n! / (n - r)! \), where \( n! \) is the factorial of n, which is the product of all positive integers up to n.

Step 2: Applying the formula to our problem

In this scenario, both n and r are equal to 6, as we have six people and six seats. Hence, we need to calculate \( 6P6 \), which equals \( 6! / (6 - 6)! = 6! / 0! \). Any non-zero number's factorial is the product of all positive numbers less than or equal to that number, and \( 0! \) is defined as 1.

Step 3: Calculating the answer

\( 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720 \), and since \(0! = 1\), we have \(6P6 = 6! / 0! = 720 / 1 = 720\).