Problem 66 - College Algebra

Question

Write a word problem that can be solved by evaluating \(_{7} C_{3}\)

Step-by-Step Solution

Answer

There are 35 different ways the coach can select 3 players out of 7 for a special training session.

Step 1: Context Selection

Let's choose a context in which this problem can apply. A good example is a group of people or objects from which a smaller group needs to be selected. For instance, we can use the following context: 'There are 7 players in a basketball team and the coach needs to select 3 players for a special training session.'

Step 2: Formulate the Problem

Now we can formulate the problem. The best way to do this is by asking a question that involves the selection of the smaller group. For instance: 'In how many ways can the coach select 3 players for a special training session from a team comprising of 7 players?' This problem will require the evaluation of \(_{7} C_{3}\) to find the solution.

Step 3: Evaluate the Combination

The number of ways the coach can select these 3 players is calculated using the combination formula, \(_{n} C_{r}\) which is given by \(\frac{n!}{r!(n-r)!}\). Substituting \(n=7\) and \(r=3\), we get \(_{7} C_{3} = \frac{7!}{3!(7-3)!} = \frac{7*6*5}{3*2*1} = 35\) different ways.

Key Concepts

Combinations

In mathematics, combinations are a key concept used in counting and probability, focusing on the selection of items from a larger set where the order does not matter. Imagine you have a basket with various fruits, and you want to choose a few to make a fruit salad. Whether you pick an apple first and then a banana, or a banana first and then an apple, the result is the same fruit salad. That's where combinations come into play.

• The combination is denoted by \( _{n} C_{r} \) where \( n \) is the total number of items, and \( r \) is the number of items to choose.
• The formula for calculating combinations is \( \frac{n!}{r!(n-r)!} \).

This is different from permutations, where the order does matter. In a permutation, picking an apple and a banana is different from picking a banana and an apple. Therefore, remember: in combinations, it's all about selection without regard to order.

Factorial Notation

Factorial notation is a powerful tool in combinatorics and is key to calculating combinations. It is used to describe the product of an integer and all the integers below it, down to 1. In simpler terms, if you see a number followed by an exclamation mark, like \( 4! \), it means 4 multiplied by 3, multiplied by 2, multiplied by 1.

• The factorial of a number \( n \) is denoted as \( n! \).
• It is calculated as \( n \times (n-1) \times (n-2) \times ... \times 1 \).
• Special case: \( 0! = 1 \).

Factorial notation simplifies the process of calculating combinations and permutations, as these formulas often involve large and complex multiplications. In our word problem, factorials help us easily determine how many ways a coach can select players from a basketball team.

Problem-Solving in Mathematics

Problem-solving in mathematics involves understanding a situation, translating it into a mathematical form, and then finding a solution. When tackling word problems, it's important to identify what you're trying to find and how you can use mathematical tools to reach an answer. Here's how problem-solving works in our basketball team scenario:

• Understand the Problem: Recognize that we need to select a certain number of players from a larger group.
• Translate to Math: Use combinations to determine the number of ways to make the selections, as order doesn't matter.
• Solve the Equation: Apply the formula \( _{n} C_{r} = \frac{n!}{r!(n-r)!} \) to determine the final solution.

Effective problem-solving skills help you not only work through exercises in a textbook but also apply these skills to real-world situations that you may encounter.

Mathematical Modeling

Mathematical modeling is the art of translating real-world problems into mathematical language so they can be analyzed and solved efficiently. In the context of our combination problem, modeling helps us frame the scenario of selecting players for a training session in a mathematical way:

• Define the Scenario: We defined a basketball team selection issue as a situation that could be modeled mathematically.
• Use Mathematical Tools: Choose the right mathematical concepts, like combinations, to represent the issue.
• Analyze and Solve: Apply the combination formula to find the solution.

Mathematical modeling is crucial in various fields, from engineering to economics, where complex systems require simple, efficient solutions. It enables us to take real-world scenarios, convert them into a problem we can solve, and then interpret the results back into practical insights.