13.5 Counting Principles

ⓐ Seven red and five green marbles are placed in a bag. How many marbles are there to choose from?

ⓑ Let the set A = {−5,−3,−1,2,3,4,5,6}. How many ways are there to choose a negative or an even number from A?

ⓒ A student is shopping for a new computer. He is deciding among 2 desktop computers and 3 laptop computers. What is the total number of computer options?

Solve counting problems using the multiplication principle.

1. ⓐ Diane packed 2 skirts, 3 blouses, and 2 sweaters for her business trip. She will need to choose a skirt and a blouse for each outfit and decide whether to wear the sweater. Use the Multiplication Principle to find the total number of possible outfits.
2. ⓑ A restaurant offers a lunch special that includes an entree, a main dish, and a b…

A student is shopping for a new computer. He is deciding among 3 desktop computers and 4 laptop computers. What is the total number of computer options?

A restaurant offers a breakfast special that includes a breakfast sandwich, a side dish, and a beverage. There are 3 types of breakfast sandwiches, 4 side dish options, and 5 beverage choices. Find the total number of possible breakfast specials.

How many ways can the family line up for the portrait?

How many ways can the photographer line up 3 family members?

How many ways can the family line up for the portrait if the parents are required to stand on each end?

A professor is creating an exam of 9 questions from a test bank of 12 questions. How many ways can she select and arrange the questions?

How many ways can the 7 actors line up?

How many ways can 5 of the 7 actors be chosen to line up?

A fast food restaurant offers five side dish options. Your meal comes with two side dishes.

1. ⓐHow many ways can you select your side dishes?
2. ⓑHow many ways can you select 3 side dishes?

An ice cream shop offers 10 flavors of ice cream. How many ways are there to choose 3 flavors for a banana split?

A restaurant offers butter, cheese, chives, and sour cream as toppings for a baked potato. How many different ways are there to order a potato?

A sundae bar at a wedding has 6 toppings to choose from. Any number of toppings can be chosen. How many different sundaes are possible?

Find the number of rearrangements of the letters in the word DISTINCT.

Find the number of rearrangements of the letters in the word CARRIER.

Assume that there are $n$ ways an event $A$ can happen, $m$ ways an event $B$ can happen, and that $A \text{and}\textrm{ } B$ are non-overlapping.

Use the Addition Principle of counting to explain how many ways event $A \text{or}\textrm{ } B$ can occur.

Answer the following questions.

When given two separate events, how do we know whether to apply the Addition Principle or the Multiplication Principle when calculating possible outcomes? What conjunctions may help to determine which operations to use?

Answer the following questions.

What is the term for the arrangement that selects $r$ objects from a set of $n$ objects when the order of the $r$ objects is not important? What is the formula for calculating the number of possible outcomes for this type of arrangement?

Determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.

Let the set $B = \left\{- 23 , - 16 , - 7 , - 2 , 20 , 36 , 48 , 72\right\} .$ How many ways are there to choose a positive or an odd number from $A ?$

Determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.

How many ways are there to pick a paint color from 5 shades of green, 4 shades of blue, or 7 shades of yellow?

Determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.

How many outcomes are possible from tossing a coin and rolling a 6-sided die?

Determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.

How many ways are there to construct a string of 3 digits if numbers can be repeated?

Compute the value of the expression.

$P \left(5 , 2\right)$

Compute the value of the expression.

$P \left(3 , 3\right)$

Compute the value of the expression.

$P \left(11 , 5\right)$

Compute the value of the expression.

$C \left(12 , 4\right)$

Compute the value of the expression.

$C \left(7 , 6\right)$

Find the number of subsets in each given set.

$\left\{1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10\right\}$

Find the number of subsets in each given set.

A set containing 5 distinct numbers, 4 distinct letters, and 3 distinct symbols

Find the number of subsets in each given set.

The set of two-digit numbers between 1 and 100 containing the digit 0

Find the distinct number of arrangements.

The letters in the word “academia”

Find the distinct number of arrangements.

The symbols in the string #,#,#,@,@,$,$,$,%,%,%,%

The set, $S$ consists of $\text{900},\text{000},\text{000}$ whole numbers, each being the same number of digits long. How many digits long is a number from $S ?$ (Hint: use the fact that a whole number cannot start with the digit 0.)

Can $C \left(n , r\right)$ ever equal $P \left(n , r\right) ?$ Explain.

How many arrangements can be made from the letters of the word “mountains” if all the vowels must form a string?

A cell phone company offers 6 different voice packages and 8 different data packages. Of those, 3 packages include both voice and data. How many ways are there to choose either voice or data, but not both?

A wholesale T-shirt company offers sizes small, medium, large, and extra-large in organic or non-organic cotton and colors white, black, gray, blue, and red. How many different T-shirts are there to choose from?

An art store has 4 brands of paint pens in 12 different colors and 3 types of ink. How many paint pens are there to choose from?

How many ways can a baseball coach arrange the order of 9 batters if there are 15 players on the team?

A motorcycle shop has 10 choppers, 6 bobbers, and 5 café racers—different types of vintage motorcycles. How many ways can the shop choose 3 choppers, 5 bobbers, and 2 café racers for a weekend showcase?

Just-For-Kicks Sneaker Company offers an online customizing service. How many ways are there to design a custom pair of Just-For-Kicks sneakers if a customer can choose from a basic shoe up to 11 customizable options?

Suni bought 20 plants to arrange along the border of her garden. How many distinct arrangements can she make if the plants are comprised of 6 tulips, 6 roses, and 8 daisies?