Problem 32 - Algebra 2 and Trigonometry

Question

In \(32-37,\) determine the number of different arrangements. The finishing order of 7 runners in a race

Step-by-Step Solution

Answer

There are 5040 different arrangements for the finishing order of 7 runners in a race.

Step 1: Understand Permutation Concept

When considering how to arrange runners in a race, we are dealing with permutations. The order in which the runners finish matters, which is the essential characteristic of permutations.

Step 2: Set up the Permutation Formula

To find the number of different arrangements of runners, we use the permutation formula: \[ P(n) = n! \]where \( n \) is the number of runners. Thus, we need to calculate \( 7! \) (7 factorial).

Step 3: Calculate the Factorial of 7

Calculate the factorial of 7 by multiplying integers from 1 to 7:\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

Step 4: Perform the Multiplication

Multiply the sequence of numbers:- \( 7 \times 6 = 42 \)- \( 42 \times 5 = 210 \)- \( 210 \times 4 = 840 \)- \( 840 \times 3 = 2520 \)- \( 2520 \times 2 = 5040 \)- \( 5040 \times 1 = 5040 \)

Step 5: Conclude the Calculation

The number of different arrangements for the finishing order of 7 runners is 5040.

Key Concepts

Factorials

In mathematics, a factorial is a special function denoted by an exclamation mark (!). It is used to find the product of all positive integers up to a specified number. For example, when you see "7!", it reads as "seven factorial". To calculate a factorial, you start with the given number and multiply it by every whole number less than itself until you reach 1. This can be written as:\[7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\]This results in 5040 in the case of 7!.

Factorials are extremely useful in permutations and combinations, as they help find the number of ways objects can be arranged or selected. Factorials grow very quickly, which is why they are so powerful in combinatorial calculations.

Arrangements

When talking about arrangements in mathematics, we're often discussing how we can order or sequence objects. In the context of this exercise, we want to determine how 7 runners can line up in a race. Each different sequence of runners finishing is considered a unique arrangement.

Understanding how to calculate arrangements is important because it highlights the significance of sequence in permutations. If you re-order the same set of items, you often get a different outcome.

• If you had 3 runners, say A, B, and C, they could finish as ABC, ACB, BAC, BCA, CAB, or CBA. Each of these is a distinct arrangement.
• With 7 runners, the factorial function helps us quickly calculate the number of possible arrangements without manually listing them.

Permutation Formula

The permutation formula is an essential mathematical tool used to determine how many different ways a set of items can be arranged. Since the order matters in permutations, it's especially useful in situations where different sequences can lead to different outcomes like race finishes.The formula for permutations when selecting and arranging all objects from a set is:\[P(n) = n!\]Here, \( n \) represents the total number of objects to arrange. So, for our exercise with 7 runners, we calculated the permutations using:\[P(7) = 7!\]This results in 5040, indicating there are 5040 different possible finishing orders for the 7 runners. The permutation formula simplifies complex arrangement problems, allowing you to find solutions quickly and effectively in a variety of real-world scenarios.