Problem 70 - Precalculus with Limits

Question

A clothing manufacturer interviews \( 12 \) people for four openings in the human resources department of the company. Five of the \( 12 \) people are women. If all \( 12 \) are qualified, in how many ways can the employer fill the four positions if (a) the selection is random and (b) exactly two women are selected?

Step-by-Step Solution

Answer

For random selection, there are 495 ways to fill the positions. For the selection of exactly two women, there are 525 ways.

Step 1: Calculate the Total Number of Combinations

If the selection is random, meaning any of the 12 people could be chosen irrespective of their gender, we can use the combination formula, which is \(C(n, k) = n! / [k!(n-k)!]\), where 'n' is the total number of people (12) and 'k' is the number of people to be selected (4). So, \(C(12, 4) = 12! / [4!(12-4)!]\).

Step 2: Calculate the Number of ways when exactly two women are to be selected

To calculate the number of ways to select exactly two women out of five and two men out of seven, we use the combination formula twice - once for the women and once for the men. So, the calculation will be \(C(5,2) * C(7,2) = 5! / [2!(5-2)!] * 7! / [2!(7-2)!]\).

Step 3: Evaluate the Factorials and Perform the Calculations

The final step is to evaluate the factorials in each of the expressions obtained in steps 1 and 2 and perform the calculations. This gives the answer to the question.

Key Concepts

Combination Formula

The combination formula is a key concept in combinatorics, often used in problems involving selections or groupings. It helps us calculate the number of ways to choose a subset of items from a larger set without considering the order.

• The formula is represented as \( C(n, k) = \frac{n!}{k!(n-k)!} \).
• Here, \( n \) is the total number of items, and \( k \) is the number you wish to select.

Imagine you have 12 interview candidates and you want to fill 4 positions. Using the combination formula, you can determine how many different sets of 4 candidates can be chosen from the pool of 12.
This exercise begins by using the formula with \( n = 12 \) and \( k = 4 \), resulting in \( C(12, 4) \), which provides all possible selections without caring about who is chosen first, second, etc.

Factorials

Factorials play a crucial role in combinatorics, particularly in the combination formula. A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a given number.
For instance, the factorial of 4 (written as \( 4! \)) is calculated as:

• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

In selection problems, factorials are utilized to determine the number of ways to arrange \( n \) items. In our interview problem, we calculate factorials to simplify the expression \( \frac{12!}{4!8!} \) when computing \( C(12, 4) \).
Understanding how to compute factorials is a fundamental skill when solving problems in combinatorics.

Selection Problems

Selection problems are a common exercise in combinatorics where you choose a subset of items from a larger group. They often involve conditions, like selecting a specific number of items that meet a certain criterion.
In the exercise, we discuss both random selection and conditional selection, using the interview scenario:

• For random selection, any of the 12 candidates can be chosen, which requires \( C(12, 4) \) combinations.
• When selecting exactly two women, the calculation involves two steps: choosing 2 women from 5 and 2 men from 7.

To solve this, apply the combination formula for each step: \( C(5, 2) \times C(7, 2) \), reflecting the separate selections. These problems not only encourage the application of formulas but also the development of logical reasoning and strategic thinking when solving real-world scenarios.