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

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1:54 minutes

Problem 3.4.34

Textbook Question

34. Lottery Number Selection A lottery has 52 numbers. In how many different ways can six of the numbers be selected? (Assume the order of selection is not important.)

Verified step by step guidance

Step 1: Recognize that this is a combination problem because the order of selection does not matter. The formula for combinations is given by:

C = \frac{n}{!}

, where n is the total number of items, r is the number of items to choose, and '!' denotes factorial.

Step 2: Identify the values of n and r from the problem. Here, n = 52 (total numbers in the lottery) and r = 6 (numbers to be selected).

Step 3: Substitute the values of n and r into the combination formula:

C = \frac{52}{!}

Step 4: Simplify the factorials in the denominator. Compute (52 - 6) = 46, so the formula becomes:

C = \frac{52}{!}

Step 5: Cancel out the common terms in the numerator and denominator. This simplifies to:

C = \frac{52 \times 51 \times 50 \times 49 \times 48 \times 47}{6}

. Compute this to find the total number of ways.

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Key Concepts

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

Combinations

Combinations refer to the selection of items from a larger set where the order does not matter. In this context, we are interested in choosing 6 numbers from a total of 52, which is a classic example of a combination problem. 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.

Factorial

A factorial, denoted as n!, is the product of all positive integers up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are essential in calculating combinations and permutations, as they help determine the total arrangements of a set of items. In the lottery question, factorials will be used to compute the number of ways to choose 6 numbers from 52.

Binomial Coefficient

The binomial coefficient, often represented as C(n, k) or 'n choose k', quantifies the number of ways to choose k elements from a set of n elements without regard to the order of selection. It is calculated using the formula C(n, k) = n! / (k!(n-k)!). This concept is crucial for solving the lottery number selection problem, as it directly provides the number of combinations of 6 numbers from 52.

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