Problem 26 - Algebra for College Students

Question

A committee of three is chosen at random from a group of five men and four women. Find the probability that the committee contains two men and one woman. \(\frac{10}{21}\)

Step-by-Step Solution

Answer

The probability is \(\frac{10}{21}\).

Step 1: Understand Total Possibilities

First, calculate the total ways to choose any 3 people from the 9 possible (5 men and 4 women). Use the combination formula \( \binom{n}{r} \), where \( n \) is the total number of people, and \( r \) is how many you choose. This is:\[ \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]Thus, there are 84 ways to select any 3 individuals from the group.

Step 2: Calculate Ways to Choose Two Men

Now, determine how many ways to choose 2 men from the 5 available. Use the combination formula:\[ \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \]This result means there are 10 ways to choose 2 men from the 5.

Step 3: Calculate Ways to Choose One Woman

Determine the ways to select 1 woman from the 4 available using the combination formula:\[ \binom{4}{1} = 4 \]Thus, there are 4 ways to choose 1 woman.

Step 4: Multiply to Find Specific Event

Multiply the number of ways to choose 2 men by the number of ways to choose 1 woman to find the number of favorable outcomes: \(10 \times 4 = 40\). This results in 40 favorable combinations where the committee contains 2 men and 1 woman.

Step 5: Calculate Probability

The probability of choosing 2 men and 1 woman from the group is the ratio of favorable outcomes to total possibilities: \( \frac{40}{84} \). Simplify this fraction by dividing the numerator and the denominator by their greatest common divisor, which is 4: \( \frac{40}{84} = \frac{10}{21} \).

Step 6: Final Answer

Based on the calculations, the probability that the committee contains exactly two men and one woman is \( \frac{10}{21} \).

Key Concepts

Combinations

In combinatorics, a combination is a selection of items from a larger pool. Importantly, the order in which these items are chosen does not matter. This is in contrast to permutations, where order does matter. A combination is useful when you want to know how many ways you can choose a specific number of elements from a set without caring about the sequence.

The formula to calculate combinations is given by the binomial coefficient, also known as "n choose r":

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

Here, \( n! \) ("n factorial") represents the product of all positive integers up to \( n \), and provides the total permutations of \( n \) items.Let's take an example similar to our original exercise. Imagine a drawer with 5 distinct pairs of socks. If you wanted to pick 2 socks without considering order, you would use the combinations formula:

• \( \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \)

Thus, there are 10 combinations possible in selecting any 2 socks from 5.

Committee Selection

Selecting a committee from a group of individuals often uses combinations, particularly when the order of selection doesn't matter. In problems like the one presented, it's not about who is chosen first, second, or last—only that certain criteria are met.When forming a committee, consider both the total group size and the specific requirements, such as gender or role. The original exercise asks for committees based on gender composition: 2 men and 1 woman. First, calculate all possible ways to form any committee from the entire group, then consider which combinations fit the specified criteria.

• Entire Group: Choose 3 out of 9 people (\( \binom{9}{3} = 84 \)).
• Specific Grouping: 2 men from 5 (\( \binom{5}{2} = 10 \)) and 1 woman from 4 (\( \binom{4}{1} = 4 \)).

After determining the possible way to select the desired subgroups (man and woman counts separately), you multiply the two combinations since each group selection is independent:

• Combinations: \( 10 \times 4 = 40 \)

This detailed method ensures a thorough understanding of how groups or committees are selected, aligning with real-world processes.

Binomial Coefficient

The binomial coefficient is a fundamental concept in combinatorics, symbolized by \( \binom{n}{r} \). It calculates the number of ways to choose \( r \) items from a total of \( n \) items, without regard to order.This mathematical tool is pivotal in probability, algebra, and calculus. It's especially crucial when solving probability problems involving selections, like forming committees or picking teams.The binomial coefficient can also be derived by simplifying factorial expressions:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

This simplifies large calculations by breaking them into more manageable multiplications and divisions. For example, in determining a committee as in our original problem, several coefficients are calculated:

• Total Selection: \( \binom{9}{3} = 84 \)
• Selecting Two Men: \( \binom{5}{2} = 10 \)
• Selecting One Woman: \( \binom{4}{1} = 4 \)

The calculations yield the answer by first determining all possible combinations, then refining to the favorable outcomes needed. This ensures an efficient and correct solution to selection-related probability problems.