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

Fundamental Counting Principle

Statistics

4. Probability

Fundamental Counting Principle

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1:10 minutes

Problem 4.4.1

Textbook Question

Notation What does the symbol ! represent? The five starting players of an NBA basketball team can stand in a line 5! different ways, so what is the actual number of ways that the five players can stand in a line?

Verified step by step guidance

The symbol '!' represents a factorial in mathematics. The factorial of a number n, denoted as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1.

To calculate the number of ways the five starting players can stand in a line, we need to compute the factorial of 5, which is written as 5!.

Write out the factorial expression for 5!: 5! = 5 × 4 × 3 × 2 × 1.

Multiply the numbers step by step: Start with 5 × 4, then multiply the result by 3, then by 2, and finally by 1.

The result of this multiplication will give you the total number of ways the five players can stand in a line.

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Key Concepts

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

Factorial

The symbol '!' represents the factorial of a number, which is the product of all positive integers up to that number. For example, 5! (read as 'five factorial') equals 5 × 4 × 3 × 2 × 1, which calculates the total number of ways to arrange five distinct objects.

Permutations

Permutations refer to the different arrangements of a set of items where the order matters. In the context of the question, the arrangement of the five NBA players in a line is a permutation, as changing the order of players results in a different arrangement.

Combinatorial Counting

Combinatorial counting is a branch of mathematics that deals with counting, arranging, and combining objects. It provides the foundational principles for calculating permutations and combinations, which are essential for solving problems involving arrangements, such as the number of ways players can line up.

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Related Practice

Textbook Question

"In Exercises 5 and 6, use the Fundamental Counting Principle.

5. A student must choose from seven classes to take at 8:00 A.M., four classes to take at 9:00 A.M., and three classes to take at 10:00 A.M. How many ways can the student arrange the schedule?"

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Multiple Choice

How many possible outcomes are there if you roll 5 dice?

How many options are there for license plates with any three letters (A-Z) followed by any 3 numbers (0-9)?

Phone numbers are 10 digits long. How many possible phone numbers are there if the 1^st and 4^th numbers can't be 0?

Soccer Shootout In the FIFA Women’s World Cup 2019, a tie at the end of two overtime periods leads to a “shootout” with five kicks taken by each team from the penalty mark. Each kick must be taken by a different player. How many ways can 5 players be selected from the 11 eligible players? For the 5 selected players, how many ways can they be designated as first, second, third, fourth, and fifth?

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Textbook Question

Jumble Many newspapers carry “Jumble,” a puzzle in which the reader must unscramble letters to form words. The letters MHRHTY were included in newspapers on the day this exercise was written. How many ways can those letters be arranged? Identify the correct unscrambling, then determine the probability of getting that result by randomly selecting one arrangement of the given letters.

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Textbook Question

Expected Value in North Carolina’s Pick 4 Game In North Carolina’s Pick 4 lottery game, you can pay $1 to select a four-digit number from 0000 through 9999. If you select the same sequence of four digits that are drawn, you win and collect $5000.

a. How many different selections are possible?

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Textbook Question

Identifying the Sample Space of a Probability Experiment In Exercises 25-32, identify the sample space of the probability experiment and determine the number of outcomes in the sample space. Draw a tree diagram when appropriate.

31. Rolling a pair of six-sided dice

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