Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

4. Probability

Counting

Statistics

4. Probability

Counting

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2:00 minutes

Problem 3.4.50a

Textbook Question

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.

Verified step by step guidance

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 = \frac{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 = \frac{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 = \frac{200 \cdot 199 \cdot 198 \cdot 197 \cdot 196}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 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.

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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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.

Key Concepts

Combinatorics

Binomial Coefficient

Factorial

Watch next

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.