In Exercises 49-53, use counting principles to find the probability.
53. A corporation has six male senior executives and four female senior executives. Four senior executives are chosen at random to attend a technology seminar. What is the
probability of choosing
c. two men and two women?

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the probability of selecting 2 men and 2 women from a group of 6 male senior executives and 4 female senior executives, where 4 senior executives are chosen at random.
2. Step 2: Use the combination formula to calculate the number of ways to choose 2 men from the 6 male executives. The combination formula is given by: $C(n,r)=\frac{n!}{r!(n-r)!}$, where n is the total number of items, and r is the number of items to choose. Here, n = 6 and r = 2.
3. Step 3: Similarly, calculate the number of ways to choose 2 women from the 4 female executives using the same combination formula. Here, n = 4 and r = 2.
4. Step 4: Multiply the results from Step 2 and Step 3 to find the total number of favorable outcomes (choosing 2 men and 2 women). This is because the selection of men and women are independent events.
5. Step 5: Calculate the total number of ways to choose 4 executives from the entire group of 10 (6 men + 4 women) using the combination formula, where n = 10 and r = 4. Finally, divide the number of favorable outcomes (from Step 4) by the total number of outcomes (from this step) to find the probability.

Key Concepts

Counting Principles

Counting principles, such as the fundamental counting principle, permutations, and combinations, are essential for determining the number of ways to select items from a group. In this scenario, combinations are particularly relevant, as the order of selection does not matter when choosing executives. Understanding how to calculate combinations allows us to find the total number of ways to choose the required number of men and women.

Combinations

Combinations refer to the selection of items from a larger set where the order does not matter. The formula for combinations is given by C(n, k) = n! / (k!(n-k)!), where n is the total number of items, k is the number of items to choose, and '!' denotes factorial. In this problem, we will use combinations to calculate the number of ways to choose two men from six and two women from four.

Probability

Probability is a measure of the likelihood of an event occurring, expressed as a ratio of favorable outcomes to the total number of possible outcomes. To find the probability of selecting two men and two women, we will calculate the number of favorable combinations (selecting two men and two women) and divide it by the total combinations of selecting any four executives from the ten available. This gives us the desired probability.