Problem 49 - College Algebra

Question

In Exercises \(49-58,\) solve by the method of your choice. In a race in which six automobiles are entered and there are no ties, in how many ways can the first four finishers come in?

Step-by-Step Solution

Answer

The first four finishers in the race can come in 360 different ways.

Step 1: Identify n and r

In this exercise, the total number of automobiles \(n\) is six and the number of finishers \(r\) which we are interested in is four.

Step 2: Apply the permutation formula

Plug the values of \(n\) and \(r\) into the permutation formula, \(nPr = n!/(n-r)!\). This will be equal to \(6!/(6-4)!\)

Step 3: Simplify the factorial

By simplifying we get \(6! / 2!\). Recalling that \(6! = 6 * 5 * 4 * 3 * 2 * 1\) and \(2! = 2 * 1\), the calculation simplifies to \(6 * 5 * 4 * 3 = 360\)

Step 4: Make a conclusion

Hence, in a race in which six automobiles are entered and there are no ties, there are 360 ways in which the first four finishers can come in.

Key Concepts

Factorial

When dealing with permutations and combinations, understanding the concept of factorial is crucial. A factorial, denoted by an exclamation mark (!), is the product of an integer and all the positive integers below it.

For instance:

• The factorial of 3, written as 3!, is equal to: 3 x 2 x 1 = 6
• Similarly, 5! equals: 5 x 4 x 3 x 2 x 1 = 120

Basically, to find the factorial of a number, you multiply it by every number below it down to 1. This operation is essential in permutations because it helps calculate the total number of possible arrangements or sequences. It simplifies our calculations and helps manage the complexity when dealing with large numbers. Recognizing that 0! is defined to be 1 is also important. This is particularly useful when calculating combinations and permutations where the number zero may appear in calculations.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations, permutations, and the counting of configurations. It's all about figuring out how different sets of objects can be arranged or selected. In our exercise, combinatorics helps us understand different possible outcomes during a race.

In permutations, like the exercise we are analyzing, the order of arrangement is crucial. This is different from combinations where the order doesn’t matter, such as when choosing a group of friends for a team. Understanding combinatorics allows us to use tools such as the permutation formula to determine how many ways we can rearrange a set of items. By employing combinatorics, we can also explore:

• How many ways we can choose objects from a set
• The different sequences in which objects can be ordered

This broad area of mathematics provides us with the techniques necessary to compute and analyse different arrangements and possibilities in a systematic way.

Counting Principle

The counting principle is a fundamental concept used to easily determine the number of possible outcomes in various scenarios. This principle states that if one event can occur in "m" ways and another independent event can occur in "n" ways, then the two events together can occur in m * n ways.

Using this principle helps streamline complex calculations into more manageable operations. In the context of permutations, especially in our exercise involving a race, the counting principle comes into play by allowing us to determine how different cars can finish in various orders. The concept ties directly into permutations, where determining the races' outcomes boils down to arranging four out of six cars in distinctive ways. Here's how it plays out in a permutation:

• Select a car to come in first (6 choices).
• Then a car for second (5 choices after the first).
• Followed by third place (4 choices), and finally.
• The fourth position (3 choices).

Therefore, multiplying 6 x 5 x 4 x 3 gives us the 360 permutations possible as illustrated using the counting principle.