Problem 42 - College Algebra with Corequisite Support

Question

In horse racing, a "trifecta" occurs when a bettor wins by selecting the first three finishers in the exact order (1st place, 2nd place, and 3rd place). How many different trifectas are possible if there are 14 horses in a race?

Step-by-Step Solution

Answer

2184 trifectas are possible.

Step 1: Identify the Problem

We are asked to find the number of possible outcomes when choosing the first three finishers in a specific order out of 14 horses. This is a permutation problem because the order of selection matters.

Step 2: Apply the Permutation Formula

The permutation formula for choosing \( k \) items out of \( n \) items is given by \( nPk = \frac{n!}{(n-k)!} \). Here, \( n = 14 \) and \( k = 3 \). We need to calculate \( 14P3 \).

Step 3: Simplify the Formula

First, calculate \( 14! = 14 \times 13 \times 12 \times 11! \) and \( (14 - 3)! = 11! \). Thus, \[ 14P3 = \frac{14 \times 13 \times 12 \times 11!}{11!} \].

Step 4: Cancel Out the Common Factor

The \( 11! \) in the numerator and denominator cancels out, simplifying the equation to \[ 14P3 = 14 \times 13 \times 12 \].

Step 5: Calculate the Product

Multiply the remaining numbers: \( 14 \times 13 = 182 \), then \( 182 \times 12 = 2184 \).

Step 6: Finalize the Solution

Thus, the total number of different trifecta combinations in this scenario is 2184.

Key Concepts

Factorial

A factorial, denoted by the symbol "!", is a mathematical operation that involves multiplying a series of descending natural numbers. This operation is crucial in various fields of mathematics, especially in problems dealing with permutations and combinations. It helps us compute the number of possible ways to arrange a set number of items.

• The factorial of a number, say \( n \), is calculated as \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \).
• For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
• Factorials grow rapidly in size with increasing numbers, which is why they are powerful in handling large-scale permutations.

Understanding factorials is vital when dealing with permutation problems such as the trifecta in horse racing, where the arrangement and order are important.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination of objects within a set. It's particularly useful when solving problems related to permutations and combinations.
Combinatorics helps us determine:

• How many ways we can choose and arrange objects.
• When the order of arrangement matters, we use permutations. For example, choosing the top three horses in a specific order out of fourteen is a permutation.
• When the order does not matter, we turn to combinations.

In our horse racing example, since the trifecta requires the selection of horses in a specific order, combinatorics using permutations is applied to find the number of possible outcomes.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world scenario to make predictions or decisions. In the context of permutations, mathematical modeling helps us simplify complex arrangements into calculable formulas.

• By utilizing mathematical models, we apply specific formulas like the permutation formula \( nPk = \frac{n!}{(n-k)!} \) to determine possible outcomes.
• These models allow for straight-forward calculations, giving us the ability to solve otherwise complicated counting problems efficiently.
• In our horse race example, the permutation model enables us to predict the number of trifecta combinations possible from the 14 horses quickly.

Ultimately, mathematical modeling provides a structured approach to understanding and predicting outcomes from complex variables in various fields, not just in racing or sports.