Problem 72 - Algebra and Trigonometry

Question

55–75 Solve the problem using the appropriate counting principle(s). Choosing a Delegation Three delegates are to be chosen from a group of four lawyers, a priest, and three professors. In how many ways can the delegation be chosen if it must include at least one professor?

Step-by-Step Solution

Answer

There are 46 ways to choose a delegation with at least one professor.

Step 1: Calculate Total Ways Without Restriction

First, we find the total number of ways to choose any 3 delegates out of the total 8 people (4 lawyers, 1 priest, and 3 professors) without any restrictions. We use the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. Thus, the number of ways is \( \binom{8}{3} \). Calculation: \[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \] ways.

Step 2: Calculate Ways Excluding All Professors

Next, we calculate the number of ways to form a delegation with no professors. Here, we only choose from 5 people (4 lawyers and 1 priest). We need to choose 3 delegates from these 5 people. Thus, the number of ways is \( \binom{5}{3} \). Calculation: \[ \binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10 \] ways.

Step 3: Calculate Ways Including At Least One Professor

To find the number of ways that include at least one professor, we subtract the number of ways with no professors from the total number of unrestricted ways. Calculation: Total ways with at least one professor = Total ways without restriction \(-\) Ways excluding all professors \[ 56 - 10 = 46 \] ways.

Key Concepts

Combination Formula

The combination formula is a fundamental concept in combinatorics, which is used when you want to select items from a larger group without considering the order of selection. It's all about the different possible groupings. The formula is expressed as \( \binom{n}{r} \), which stands for "n choose r." Here, \( n \) is the total number of items in the group, and \( r \) is the number of items to select.

To calculate combinations, we use a specific formula:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)
• "!" denotes factorial, which is the product of an integer and all the integers below it. So, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).

This formula helps solve problems where the arrangement does not matter but finding out how many ways a selection can be made is key.

Counting Principle

The counting principle is a vital strategy in problem-solving, helping break down complex choices into manageable parts. It is based on the idea that if there are several stages and each stage can occur in several ways, you can find the total number of occurrences by multiplying the ways each stage can occur.

This principle is often employed in scenarios where combinations or permutations are necessary. It streamlines the process of determining possible outcomes by systemically counting the options available at each step. For example, if there are 3 ways to do A and 2 ways to do B, the counting principle shows there are \( 3 \times 2 = 6 \) ways to do both A and B.

The fundamental counting principle is the backbone of combinatorial reasoning, offering a simple framework to tackle more complex mathematical problems.

Permutation

Permutations are distinct from combinations because they take the order of selection into account. When the arrangement of selected items is crucial, permutations are used to calculate outcomes. The formula used is:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

Here, \( n \) is the total number of items to pick from, and \( r \) is the number of items to arrange.

• The notation emphasizes the importance of order unlike combinations where order does not impact the outcome.

Permutations play a critical role in situations where the sequence matters, such as arranging books on a shelf or scheduling appointments. Understanding the difference between permutations and combinations is crucial when solving diverse mathematical problems.

Mathematical Problem-Solving

Mathematical problem-solving is a fundamental skill that involves identifying and applying methods to tackle mathematical challenges effectively. The provided exercise is a great example of this, requiring careful application of the combination formula and understanding how to count cases accurately.

Key steps often involve:

• Defining the problem and identifying constraints.
• Choosing the relevant mathematical tools, like the counting principle or combination formula.
• Performing calculations step-by-step.
• Checking if the solution meets all given conditions.

For instance, in the exercise, solving the problem entailed using combinations to find several selections then subtracting unwanted cases to satisfy constraints. Through these processes, one not only finds the answer but gains deeper insights into approaches to tackle similar problems in the future. Ultimately, problem-solving enhances logical reasoning and analytical thinking.