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Ch. 3 - Probability
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 3 - ProbabilityProblem 3.4.50a
Chapter 3, Problem 3.4.50a

50. Investment Committee A company has 200 employees, consisting of 144 women and 56 men. The company wants to select five employees to serve as an investment committee.
a. Use technology to find the number of ways that 5 employees can be selected from 200.

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Step 1: Recognize that this is a combination problem because the order in which the employees are selected does not matter. The formula for combinations is given by: C=n!r!(n-r)!, where n is the total number of items (employees) and r is the number of items to be selected (committee members).
Step 2: Substitute the given values into the formula. Here, n is 200 (total employees) and r is 5 (committee members). The formula becomes: C=200!5!(200-5)!.
Step 3: Simplify the factorials in the formula. Note that 200! is the product of all integers from 1 to 200, but you only need to calculate up to the first 5 terms because the rest will cancel out with the denominator. The simplified formula becomes: C=200⋅199⋅198⋅197⋅1965⋅4⋅3⋅2⋅1.
Step 4: Use technology (such as a calculator or software like Excel, Python, or a statistical tool) to compute the numerator and denominator separately. Divide the numerator by the denominator to find the total number of combinations.
Step 5: Interpret the result as the total number of ways to select 5 employees from a group of 200. This value represents the number of unique committees that can be formed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and permutations of objects. It provides the tools to count the number of ways to select items from a larger set without regard to the order of selection. In this context, we need to calculate the number of ways to choose 5 employees from a total of 200, which is a classic combinatorial problem.

Binomial Coefficient

The binomial coefficient, often denoted as C(n, k) or 'n choose k', represents the number of ways to choose k elements from a set of n elements without considering the order. It is calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial. This concept is essential for solving the problem of selecting 5 employees from 200.
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Factorial

A factorial, denoted as n!, is the product of all positive integers up to n. It is a fundamental concept in combinatorics used to calculate permutations and combinations. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Understanding factorials is crucial for applying the binomial coefficient formula to determine the number of ways to select employees.
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