Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Horserace results Ten horses are entered in a race. If the possibility of a tie for any place is ignored, in how many ways can the first-, second-, and third-place winners be determined?

Step-by-Step Solution

Answer

There are 720 ways to determine the first, second, and third place winners.

Step 1: Understanding the Problem

We have 10 horses, and we need to determine how many ways we can choose first, second, and third place winners. Importantly, order matters, and we cannot have a tie.

Step 2: Identify the Mathematical Method

This is a permutation problem because we are arranging 3 positions (first, second, third) from a total of 10 horses, and the order of arrangement is essential.

Step 3: Apply the Permutation Formula

The permutation formula for selecting k items from n items is given by: \[ P(n, k) = \frac{n!}{(n-k)!} \]. In this scenario, \( n = 10 \) and \( k = 3 \).

Step 4: Calculate the Permutation

Substitute the values into the permutation formula: \[ P(10, 3) = \frac{10!}{(10-3)!} = \frac{10 \times 9 \times 8 \times 7!}{7!} = 10 \times 9 \times 8 \].

Step 5: Perform the Multiplication

Calculate: \( 10 \times 9 = 90 \) and then \( 90 \times 8 = 720 \).

Step 6: Conclusion

Therefore, there are 720 different ways to arrange the first three places among 10 horses.

Key Concepts

Permutation formula

Permutations are a fundamental concept in mathematics, often used to calculate the number of ways in which a certain set of elements can be arranged, given that the order of arrangement is important. In a permutation scenario, we're usually dealing with a particular number of positions that need to be filled from a larger set of possibilities. The permutation formula is given by: \[ P(n, k) = \frac{n!}{(n-k)!} \] where:

• \( P(n, k) \) represents the number of permutations of \( n \) items taken \( k \) at a time.
• \( n! \) ("n factorial") denotes the product of all positive integers up to \( n \).
• \( (n-k)! \) denotes the product of all positive integers up to \( n-k \).

This formula helps calculate the number of possible ordered arrangements. Understanding how to use this formula is crucial when solving problems where arrangement plays a key role, like determining positions in a race.

Horse race

Imagine a horse race with numerous competitors. In this exercise, we have 10 horses racing for the top three places. When indulging in this classic scenario, it is vital to realize every placement matters. Being in first is very different from being in third. Thus, we have to carefully determine each distinguishable finishing order for the top three positions.

A horse race is a great example to illustrate permutations. Because every place—first, second, third—represents a unique outcome, it distinguishes between different sequences of placement among horses. These sequences rely on using the permutation formula, ensuring we account for every channel through which these horses might finish—a concept pivotal in grasping permutations.

Combinatorial arrangements

Combinatorial arrangements explore the variety of ways in which elements can be ordered or combined. Though often confused with combinations, where order does not matter, permutations focus on situations where the order of elements is paramount.
To visualize, think of these arrangements like puzzles where each piece must fit perfectly in a specific order.
In the horse race context, combinatorial arrangements could mean determining all possible ways to assign each horse a position from the first to the third place—as we've identified, an exercise best achieved using permutations.
The beauty of these arrangements lies in their assembly: comprehending how individual decisions (or choices) build into larger structured sequences.

Order matters in permutations

A distinctive trait of permutations is that the order of items makes a big difference. Unlike combinations, every different sequence counts as a different permutation since arrangements aren't identical just because the same items are used.
In permutations, think of each sequence as a new set, rearranging elements supplies new meanings and outcomes. In the horse race example, having Horse A in the first place and Horse B in the second is entirely distinct from having Horse B in the first and Horse A in the second. These sequences highlight the importance of order. Mathematically, factoring in the order seen in permutations expands the number of outcomes significantly, making it an essential computation for events like races, seating plans, or any scenario where sequential arrangement is a factor.