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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1:18 minutes

Problem 3.4.6

Textbook Question

True or False? In Exercises 3-6, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
6. 7C5=7C2

Verified step by step guidance

Step 1: Recall the formula for combinations, which is used to calculate the number of ways to choose r items from a set of n items. The formula is:

C (n, r) = \frac{n!}{r! (n - r)!}

Step 2: Apply the formula to calculate

C_{7} 5

. Substitute n = 7 and r = 5 into the formula:

\frac{7!}{5! (7 - 5)!}

Step 3: Simplify the denominator of the formula for

C_{7} 5

. The denominator becomes

5! (2!)

, since

7 - 5 = 2

Step 4: Similarly, calculate

C_{7} 2

using the same formula. Substitute n = 7 and r = 2 into the formula:

\frac{7!}{2! (7 - 2)!}

Step 5: Observe that the calculations for

C_{7} 5

and

C_{7} 2

are equivalent because of the symmetry property of combinations:

C_{n} r = C_{n} (n - r)

. Conclude that the statement is true.

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Key Concepts

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

Combinatorial Notation

Combinatorial notation, often represented as nCr or C(n, r), denotes the number of ways to choose r elements from a set of n elements without regard to the order of selection. It is calculated using the formula nCr = n! / (r!(n-r)!), where '!' denotes factorial, the product of all positive integers up to that number.

Factorial

A factorial, denoted as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are fundamental in combinatorics, particularly in calculating combinations and permutations, as they help determine the total arrangements or selections possible.

Properties of Combinations

One important property of combinations is that C(n, r) = C(n, n-r). This means that choosing r elements from n is equivalent to leaving out n-r elements. This property is crucial for simplifying combinatorial expressions and understanding relationships between different combinations.

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Related Practice

Textbook Question

Pick 10 Lottery For the New York Pick 10 lottery, the player first selects 10 numbers from 1 to 80. Then there is an official drawing of 20 numbers from 1 to 80. The prize of $500,000 is won if the 10 numbers selected by the player are all included in the 20 numbers that are drawn. Find the probability of winning that prize.

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Textbook Question

Matching Probabilities In Exercises 11-16, match the event with its probability.

a. 0.95

b. 0.005

c. 0.25

d. 0

e. 0.375

f. 0.5

16. You toss a coin four times. What is the probability of tossing tails exactly half of the time?

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Textbook Question

1. When you calculate the number of permutations of n distinct objects taken r at a time, what are you counting? Give an example.

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Textbook Question

True or False? In Exercises 3-6, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

3. A combination is an ordered arrangement of objects.

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Textbook Question

In Exercises 7-14, perform the indicated calculation.

9.8C3

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Textbook Question

In Exercises 7-14, perform the indicated calculation.

12. (10C7)/(14C7)

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Textbook Question

In Exercises 15-18, determine whether the situation involves permutations, combinations, or neither. Explain your reasoning.

15. The number of ways 16 floats can line up in a row for a parade

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Textbook Question

In Exercises 15-18, determine whether the situation involves permutations, combinations, or neither. Explain your reasoning.

17. The number of ways 2 captains can be chosen from 28 players on a lacrosse team

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True or False? In Exercises 3-6, determine whether the statement is true or false. If it is false, rewrite it as a true statement.6. 7C5=7C2

Key Concepts

Combinatorial Notation

Factorial

Properties of Combinations

Watch next

True or False? In Exercises 3-6, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
6. 7C5=7C2