Probability

Module 8: Probability Counting Principles

Combinations

Lesson

Using Combinations to Count

For some counting problems, order is not important. For instance, in most card games the order in which y our cards are dealt is not important. These unordered groupings are called combinations. A combination is a selection of r objects f rom a group of n objects where the order is not important. Combinations are denoted by nCr, and is

sometimes read "n choose r." Problems inv olv ing the number of combinations may hav e answers that are v ery large. In numbers with f our digits or more, make sure to put commas ev ery third digit f rom the right. For example: 15,600,200.

The number of combinations of r objects taken f rom a group of n objects can be f ound using the f ormula nCr = .

Example:

A swim team has 12 swimmers who can swim in a f reesty le ev ent. Find the number of possible f our-person teams chosen f rom the eligible swimmers. In this case, the order is NOT important. Use the f ormula f or combinations and let n = 12 and r = 4.

Solution:

nCr =

There are 495 4-person teams.

Example:

You are president of a club at y our school. Your sponsor giv es y ou the task of ordering pizza f or a meeting that will be held this af ternoon af ter school. The pizza shop that y ou are ordering f rom of f ers twelv e dif f erent toppings to choose f rom. You want to order three-topping pizzas. You need to determine how many pizzas can be f ormed that meet the criteria so that y ou can call and place y our order in time f or the meeting. While thinking about the dif f erent combinations of toppings, y ou realize that the order that y ou select the toppings is not important. That means that choosing pepperoni, sausage, and then mushrooms is the same as choosing sausage, pepperoni, and then mushrooms. The pizza will still contain the same three toppings.

Solution:

Since y ou know that y ou are choosing 3 toppings per pizza f rom 12 total toppings, that means that y ou are going to calculate using this f ormula:

12C3 =

There are 220 possible 3-topping pizzas.

Using the Addition and Multiplication Principles

When f inding the number of way s both ev ent A and an ev ent B can occur, y ou need to multiply. When f inding the number of way s that ev ent A or ev ent B can occur, y ou need to add.

Example: A restaurant serv es omelets that can be ordered with any of the ingredients shown:

a. Suppose y ou want exactly two v egetarian ingredients and 1 meat ingredient in y our omelet. How many dif f erent ty pes of omelets can y ou order?

b. Suppose y ou can af f ord at most 3 ingredients in y our omelet. How many dif f erent ty pes of omelets can y ou order?

Solution:

a- You need to choose two v egetarian ingredients f rom the six possible AND one meat ingredient f rom the f our possible. Multiply 6C2 * 4C1

6C2 =

4C1 =

6C2 * 4C1 = 15 * 4 = 60

There are 60 possible omelets with exactly two v egetarian and one meat ingredient.

b- You want to choose an omelet with either 0, 1, 2, or 3 ingredients. It does not matter if they are v egetarian or meat ingredients, so there are 10 to choose f rom. Add 10C0 + 10C1 + 10C2 + 10C3

10C0 =

10C1 =

10C2 =

10C3 =

10C0 + 10C1 + 10C2 + 10C3 = 1 + 10 + 45 + 120 = 176

There are 176 omelettes with at most 3 ingredients.

Combinations https://cobbk12.blackboard.com/bbcswebdav/institution/eHigh School/C...

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Finding Probabilities Using Combinations

Probability is the number of f av orable outcomes div ided by the total number of outcomes. You may need to use combinations to determine the numbers.

Example:

There are 28 students in y our math class. Your teacher chooses 5 students at random to complete a group project. Find the probability that y ou and y our best f riend are chosen to work in the group.

Solution:

Number of f av orable outcomes: If y ou AND y our best f riend are chosen, that means that three other students must be chosen f rom the remaining 26 av ailable students.

26C3 =

28C5 =

P(y ou and y our f riend are chosen) =

Self-Check

Self-Check Problem 1

For y our school pictures, y ou can choose 4 backgrounds f rom a list of 10, how many combinations of backdrops are possible?

C heck A nswer

Self-Check Problem 2

A school's student council has 16 members, including 4 seniors. There are 4 members randomly chosen to represent the student council at a school open house. What is the

probability that all 4 council members chosen are seniors?

a.

b.

C heck A nswer

Self-Check Problem 3

There are 28 students in y our math class. Your teacher chooses 5 students at random to complete a group project. Find the probability that y ou and y our best f riend are chosen

to work in the group.

a.

b.

c.

C heck A nswer

Combinations https://cobbk12.blackboard.com/bbcswebdav/institution/eHigh School/C...

2 of 2 11/28/2018, 12:08 PM