Problem 37 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In how many ways can a member of a hiring committee select 3 of 12 job applicants for further consideration?

Step-by-Step Solution

Answer

There are 220 ways that a member of the hiring committee can select 3 out of 12 job applicants for further consideration.

Step 1: Write down the given information

We are given that the total number of job applicants is 12 (n = 12) and we need to select 3 of them for further consideration (r = 3).

Step 2: Use the combination formula to find the answer

We can find the number of ways to select 3 out of 12 applicants using the combination formula: \(C(12, 3) = \frac{12!}{3! (12-3)!}\)

Step 3: Perform the calculations

Let's calculate the factorials first: 12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 3! = 3 × 2 × 1 (12-3)! = 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 Now, let's simplify the combination formula: \(C(12, 3) = \frac{12!}{3! \cdot 9!} = \frac{12 × 11 × 10 × 9!}{3! \cdot 9!}\) Notice that 9! can be canceled from both the numerator and denominator: \(C(12, 3) = \frac{12 × 11 × 10}{3!}\) Now let's calculate the remaining values: \(C(12, 3) = \frac{12 × 11 × 10}{3 × 2 × 1} = 220\)

Step 4: State the final answer

There are 220 ways that a member of the hiring committee can select 3 out of 12 job applicants for further consideration.

Key Concepts

Factorials

The concept of factorials is fundamental in combinatorics. A factorial, denoted by an exclamation mark (\(!\)), represents the product of all positive integers up to a certain number. It’s a way to describe the number of ways to arrange \(\textit{n}\) distinct objects.

• For example, \(n!\) means multiply all whole numbers from \(n\) down to 1.
• So, \(5!\) would be \(5 \times 4 \times 3 \times 2 \times 1 = 120\).

Factorials grow fast with larger numbers, which makes them very useful in permutation and combination calculations. In the case of the exercise, the factorials are used to calculate the number of combinations of applicants selected from a total group. Factorials simplify these calculations and help determine the various possible outcomes associated with permutations and combinations.

Permutations

Permutations refer to all possible ways to arrange a set of objects. It's important when the order of selection is considered. Unlike combinations where the order doesn’t matter, permutations are about "ordering" items.

• The formula for permutations of \(n\) items taken \(r\) at a time is \(P(n, r) = \frac{n!}{(n-r)!}\).
• This formula reflects all the different ways to choose items where sequence matters.

For example, if you have 3 distinct letters and you want to know the number of ways to arrange 2 of them, permutations would help determine that. Although the original exercise uses combinations, understanding permutations helps clarify what changes when ordering becomes essential. Both concepts are useful tools in discrete mathematics.

Discrete Mathematics

Discrete Mathematics is the backbone of computer science and data studies; it focuses on distinct and separate objects. It's the study of mathematical structures that are fundamentally discrete, not supporting or requiring the notion of continuity.

• Topics include logic, set theory, and graph theory.
• It plays a crucial role in solving problems related to algorithms, networking, and cryptography.

Combinatorics, which deals with combinations and permutations, is a branch of discrete mathematics. The principles explored in the exercise are examples of discrete math applied to real-world scenarios. By using combinations, discrete mathematics helps us solve problems involving a discrete set of elements, such as choosing applicants for job interviews. It lays the critical foundation for decision-making processes applied in computing and daily problem-solving.