Combinatorics and Counting Principles – Liberal Arts Math Study Guide

Q1. A restaurant offers 7 appetizers and 11 main courses. In how many ways can a person order a two-course meal?

Background

Topic: Fundamental Principle of Counting (Multiplication Principle)

This question tests your understanding of how to count the number of possible outcomes when making independent choices.

Key Terms and Formulas

• Multiplication Principle: If one event can occur in $m$ ways and a second event can occur in $n$ ways, then the two events together can occur in $m \times n$ ways.

Step-by-Step Guidance

1. Identify the number of choices for each course: 7 appetizers and 11 main courses.

2. Apply the multiplication principle to find the total number of two-course meal combinations.

3. Set up the multiplication: $7 \times 11$.

Try solving on your own before revealing the answer!

Final Answer: 77 ways

There are $7 \times 11 = 77$ possible two-course meal combinations.

Q2. A popular brand of pen is available in 6 colors and 2 writing tips. How many different choices of pens do you have with this brand?

Background

Topic: Counting Principle (Multiplication Principle)

This question is about determining the number of possible combinations when choosing from two independent options.

Key Terms and Formulas

• Multiplication Principle: Total choices = (number of colors) $\times$ (number of tips)

Step-by-Step Guidance

1. List the number of options for each feature: 6 colors and 2 tips.

2. Multiply the number of choices for each feature to find the total number of pen combinations.

3. Set up the multiplication: $6 \times 2$.

Try solving on your own before revealing the answer!

Final Answer: 12 choices

There are $6 \times 2 = 12$ different pen choices available.

Q3. A sales representative can take one of 3 different routes from City C to City F and any one of 7 different routes from City F to City M. How many different routes can she take from City C to City M, going through City F?

Background

Topic: Counting Principle (Multiplication Principle)

This question asks you to count the number of possible paths when each leg of a journey has independent options.

Key Terms and Formulas

• Multiplication Principle: Total routes = (routes from C to F) $\times$ (routes from F to M)

Step-by-Step Guidance

1. Identify the number of route options for each segment: 3 from C to F, 7 from F to M.

2. Multiply the number of options for each segment to find the total number of routes.

3. Set up the multiplication: $3 \times 7$.

Try solving on your own before revealing the answer!

Final Answer: 21 routes

There are $3 \times 7 = 21$ different possible routes from City C to City M via City F.

Q4. In a different plan for area codes, the first digit could be any number from 0 through 5, the second digit was either 7 or 8, and the third digit could be any number except 5. With this plan, how many different area codes are possible?

Background

Topic: Counting Principle (Multiplication Principle)

This question involves counting the number of possible codes given restrictions on each digit.

Key Terms and Formulas

• Multiplication Principle: Total codes = (choices for first digit) $\times$ (choices for second digit) $\times$ (choices for third digit)

Step-by-Step Guidance

1. Count the number of choices for each digit:

   • First digit: numbers 0 through 5 (how many options is that?)

   • Second digit: 7 or 8 (how many options?)

   • Third digit: any number except 5 (how many options from 0-9, excluding 5?)

2. Multiply the number of choices for each digit together.

3. Set up the multiplication: (number of first digit options) $\times$ (number of second digit options) $\times$ (number of third digit options).

Try solving on your own before revealing the answer!

Final Answer: 108 area codes

First digit: 6 options (0-5). Second digit: 2 options (7 or 8). Third digit: 9 options (0-9 except 5). $6 \times 2 \times 9 = 108$ area codes.

Q5. There are six performers who will perform their comedy acts this weekend at a comedy club. How many different ways are there to schedule their appearances?

Background

Topic: Permutations

This question is about finding the number of ways to arrange a set of distinct items (performers) in order.

Key Terms and Formulas

• Permutation: The number of ways to arrange $n$ distinct objects is $n!$ (n factorial).

• $n! = n \times (n-1) \times (n-2) \times \ldots \times 1$

Step-by-Step Guidance

1. Identify the total number of performers: 6.

2. Since all performers are to be arranged in order, use the permutation formula for all 6: $6!$.

3. Set up the factorial: $6 \times 5 \times 4 \times 3 \times 2 \times 1$.

Try solving on your own before revealing the answer!

Final Answer: 720 ways

There are $6! = 720$ different ways to schedule the performers.

Q6. There are 7 performers who will present their acts at a talent show this weekend. One of the performers insists on going last. If this performer’s request is granted, how many different ways are there to schedule the appearances?

Background

Topic: Permutations with Restrictions

This question tests your ability to count arrangements when one item is fixed in a specific position.

Key Terms and Formulas

• Permutation: $n!$ ways to arrange $n$ objects.

• If one object is fixed, arrange the remaining $(n-1)$ objects: $(n-1)!$

Step-by-Step Guidance

1. Fix the performer who insists on going last in the final position.

2. Arrange the remaining 6 performers in the other positions: $6!$ ways.

3. Set up the factorial: $6 \times 5 \times 4 \times 3 \times 2 \times 1$.

Try solving on your own before revealing the answer!

Final Answer: 720 ways

With one performer fixed last, there are $6! = 720$ ways to arrange the rest.

Q7. You need to arrange five of your favorite books along a small shelf. How many different ways can you arrange the books, assuming the order of the books makes a difference to you?

Background

Topic: Permutations

This question is about finding the number of ways to arrange a set of distinct objects in order.

Key Terms and Formulas

• Permutation: $n!$ ways to arrange $n$ objects.

Step-by-Step Guidance

1. Identify the total number of books: 5.

2. Since the order matters, use the permutation formula: $5!$.

3. Set up the factorial: $5 \times 4 \times 3 \times 2 \times 1$.

Try solving on your own before revealing the answer!

Final Answer: 120 ways

There are $5! = 120$ different ways to arrange the books.

Q8. A camp counselor and six campers are to be seated along a picnic bench. In how many ways can this be done if the counselor must be seated in the third seat and a camper who has a tendency to engage in food fights must sit to the counselor’s immediate right?

Background

Topic: Permutations with Multiple Restrictions

This question involves arranging people with specific seating requirements.

Key Terms and Formulas

• Permutation: $n!$ ways to arrange $n$ objects.

• When certain positions are fixed, arrange the remaining objects in the remaining seats.

Step-by-Step Guidance

1. Fix the counselor in the third seat.

2. Fix the food-fight camper in the fourth seat (immediately to the right of the counselor).

3. Arrange the remaining 5 campers in the other 5 seats (positions 1, 2, 5, 6, 7): $5!$ ways.

4. Set up the factorial: $5 \times 4 \times 3 \times 2 \times 1$.

Try solving on your own before revealing the answer!

Final Answer: 120 ways

With the counselor and food-fight camper fixed, there are $5! = 120$ ways to arrange the remaining campers.