Problem 24 - Algebra for College Students

Question

In how many different ways can a Mercedes, a Cadillac, and a Ford be awarded to 3 people chosen from the 9 finalists in a contest?

Step-by-Step Solution

Answer

There are 504 different ways to award the cars.

Step 1: Understand the problem

We need to find the number of ways to award 3 different cars (Mercedes, Cadillac, and Ford) to 3 out of 9 finalists. This is a permutations problem since the order in which the cars are awarded matters.

Step 2: Choose 3 finalists from 9

First, determine how many ways we can choose 3 finalists out of the 9 contestants. This is given by the combination formula: \(\binom{9}{3} = \frac{9!}{3! \times (9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84\). Note: The exclamation mark (!) denotes factorial, which means the product of all positive integers up to that number.

Step 3: Assign cars to the chosen finalists

Once we have chosen the 3 finalists, we need to assign each of the 3 cars to one of these finalists. This can be done in \(3! = 3 \times 2 \times 1 = 6\) ways.

Step 4: Calculate the total number of ways

Multiply the number of ways to choose the finalists by the number of ways to assign the cars: \(\binom{9}{3} \times 3! = 84 \times 6 = 504\).

Key Concepts

Factorial

The factorial of a number is the product of all positive integers up to that number. It is denoted by an exclamation mark (!) after the number. For example, the factorial of 5 (written as 5!) is calculated as follows:
5! = 5 × 4 × 3 × 2 × 1 = 120.
Factorials are crucial in permutations and combinations as they help calculate the number of ways to arrange or choose items.
Here’s a quick look at some factorials:

• 3! = 3 × 2 × 1 = 6
• 4! = 4 × 3 × 2 × 1 = 24
• 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

Factorials grow very quickly with larger numbers, which makes them incredibly useful in combinatorial calculations.

Combination Formula

The combination formula is used to determine the number of ways to choose a subset of items from a larger set when the order does not matter. The formula is given by:
\(\binom{n}{k} = \frac{n!}{k! (n-k)!} \)
where \( n\) is the total number of items, and \( k\) is the number of items to choose.
For instance, in the exercise, we needed to choose 3 finalists out of 9 contestants. We applied the combination formula as:
\( \binom{9}{3} = \frac{9!}{3! (9-3)!} = \frac{9 × 8 × 7}{3 × 2 × 1} = 84 \)
This tells us there are 84 different ways to choose 3 finalists from the 9 contestants.

Permutations

Permutations refer to the different ways items can be arranged where the order matters. When selecting and arranging items, the number of permutations is critical.
For the given problem, once 3 finalists are chosen out of 9, we need to assign 3 different cars to these 3 finalists. Each of the finalists can receive a different car; hence, the arrangement matters.
The number of permutations of 3 items (cars) is found using the factorial concept:
\(3! = 3 × 2 × 1 = 6\)
This means there are 6 unique ways to assign the 3 cars to the 3 finalists. Finally, we multiply this number by the number of combinations to find the total number of ways:
\( \binom{9}{3} × 3! = 84 × 6 = 504 \)
Thus, there are 504 different ways to award the 3 cars to the 3 chosen finalists.