Problem 70 - Algebra and Trigonometry

Question

Committee Membership A mathematics department consists of ten men and eight women. Six mathematics faculty members are to be selected at random for the curriculum committee. (a) What is the probability that two women and four men are selected? (b) What is the probability that two or fewer women are selected? (c) What is the probability that more than two women are selected?

Step-by-Step Solution

Answer

Compute probabilities using combinations; total ways is 18C6.

Step 1: Understanding the Problem

We have a department with 10 men and 8 women. We need to select 6 members. We will calculate probabilities for selecting certain combinations of men and women.

Step 2: Calculate Total Ways to Select Committee Members

First, calculate the total number of ways to choose 6 committee members from 18 faculty members (10 men + 8 women) using the combination formula: \[ \binom{18}{6} = \frac{18!}{6!(18-6)!} \]

Key Concepts

Combinatorics

Combinatorics is a fascinating area of mathematics that deals with counting, arrangement, and combination of objects. It's like solving puzzles with numbers and possibilities. The main elements of combinatorics include permutations and combinations. While permutations are concerned with arranging objects in specific orders, combinations focus on selecting objects without regard to the order. This branch of mathematics helps us solve problems related to probability, like the one in our exercise.

• Permutations: Order matters
• Combinations: Order does not matter

In the context of choosing faculty members for a committee, we use combinations because the order in which we select members does not impact the outcome.

Combination Formula

The combination formula allows us to determine how many ways we can select a certain number of items from a larger pool. This is crucial when order is not important, just like in our faculty selection example. The formula is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where:

• \(n\) is the total number of items to choose from
• \(r\) is the number of items to select
• \(!\) denotes a factorial, which means multiplying a series of descending natural numbers

A factorial, represented by the symbol \(!\), means that you keep multiplying the current number by each number less than it down to 1. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
Thus, in our problem, when choosing 6 faculty members from a total of 18, we use this formula to calculate the different ways to create a committee.

Faculty Selection

Selecting faculty members for a committee can be thought of as a real-world application of combinations. In our scenario, we have a mathematics department, which consists of 10 men and 8 women. The task is to form a curriculum committee composed of 6 faculty members. The challenge lies in calculating the number of ways to form a committee with specific gender compositions.
There are three probabilities to determine:

• The probability of selecting exactly 2 women and 4 men
• The probability of selecting 2 or fewer women
• The probability of selecting more than 2 women

To find these probabilities, we use the combination formula to calculate the number of favorable outcomes, then divide by the total number of possible combinations.

Mathematics Department

In the context of our exercise, the mathematics department serves as the pool from which we draw members to form a committee. We have 18 potential members: 10 men and 8 women. These members represent the possible "items" for our combinations.

• 10 male faculty members
• 8 female faculty members

Understanding the composition of the department is crucial, as it directly influences the combination calculations we make. This ensures we account for all potential groupings based on gender for forming the committee. The department setting allows us to explore combinatorial problems that involve probabilities and informs us how probabilities can be calculated in practical scenarios.