Problem 31 - Algebra and Trigonometry

Question

These problems involve permutations. Running a Race In how many different ways can a race with nine runners be completed, assuming that there is no tie?

Step-by-Step Solution

Answer

There are 362,880 different ways the nine runners can finish the race.

Step 1: Understand the Concept of Permutations

A permutation is an arrangement of objects in a specific order. For this problem, we must arrange 9 runners in different positions from 1st to 9th in the race. Since there are no ties, each runner will have a unique position.

Step 2: Identify the Number of Objects to Arrange

There are 9 runners in the race. Hence, we need to arrange 9 objects, which are the 9 runners, in 9 positions.

Step 3: Apply the Permutation Formula

The formula for permutations when arranging all objects is given by the factorial of the number of objects. For 9 objects, the permutation is given by \(9!\).

Step 4: Calculate the Factorial

Calculate \(9!\), which means multiplying all whole numbers from 9 down to 1: \(9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\).

Step 5: Perform the Calculation

Calculate \(9!\): \(9 \times 8 = 72\), \(72 \times 7 = 504\), \(504 \times 6 = 3024\), \(3024 \times 5 = 15120\), \(15120 \times 4 = 60480\), \(60480 \times 3 = 181440\), \(181440 \times 2 = 362880\), and finally, \(362880 \times 1 = 362880\).

Step 6: Conclude the Solution

The number of different ways the 9 runners can complete the race is \(362880\).

Key Concepts

Understanding Factorials

Factorials are a fundamental concept in mathematics, especially when dealing with permutations and combinations. The symbol for a factorial is an exclamation mark (!). When you see a number followed by an exclamation mark, like 9!, it means you need to multiply that number by every positive whole number less than itself. So, for 9!, you calculate it as follows:

• Start with 9, then multiply by 8
• Continue multiplying by every whole number down to 1

In mathematical terms, the factorial of a number n is expressed as: \[ n! = n \times (n-1) \times (n-2) \times ... \times 1 \]This operation provides the number of ways to arrange n distinct items in a sequence. In our race example, 9! gives us the possible permutations of arranging 9 runners with no ties. This comes out to a total of 362,880 different arrangements! Using factorials is crucial in understanding how many different ways events can take place, especially where order matters.

Basics of Combinatorics

Combinatorics is the branch of mathematics focusing on counting, arrangement, and combination of elements. It helps us solve problems related to how objects are chosen and arranged. In simpler terms, it's about figuring out combinations and permutations of sets and subsets.

Permutations vs. Combinations

• Permutations: Order matters, and you are arranging items or elements.
• Combinations: Order doesn't matter, and you are selecting groups.

Understanding the distinction between permutations and combinations can guide you through various problems. In our case, since runners finishing in different order gives different outcomes, permutations are used.

Combinatorics is essential in fields like probability, statistics, computer science, and much more, providing a way to systematically and efficiently deal with complex counting scenarios.

Importance of Arrangements

Arrangements refer to the different ways that items can be ordered or organized. In permutation problems, the order of these arrangements is crucial.

When we compute permutations, we're interested in how the sequence of elements changes the result. This is different from just selecting elements, where order might not be significant.

• An arrangement involving all elements is represented by a factorial of the total number of items.
• For a subset arrangement, the permutations formula is applied differently.

In practical situations such as races, seating plans, or schedules, understanding permutations helps in planning and decision-making. In our race example, each specific sequence of runner finishes is unique, forming a distinct arrangement. Thus, arrangements and permutations are keys to understanding and solving many real-world counting problems.