F Ma 12 Name:

4.5 Exploring Combinations

Big Idea:

Probabilistic thinking informs decision making in situations involving chance and uncertainty.

Curricular Competencies:

Explore, analyze and apply mathematical ideas using reason

Develop, demonstrate and apply conceptual understanding of mathematical ideas

Goal:

Explore how counting combinations differs from counting permutations.

Combinations

As part of the Grade 12 English course, students are required to read the following three books in a three month period: The Grapes of Wrath, The Wars, The Bean Trees

a. Due to previous late returns, Carl is only allowed to sign out one English book from the library at a time. List all of the different orders in which Carl could sign out the three books.

b. Devon is allowed to sign out all three books at the same time. How many different ways can he sign out all three books at the same time?

A is an example of a where the is

B is an example of a where the is important.

Time is getting short and the English teacher has adjusted the required reading to only two of the three books. What are the number of ways that Carl and Devon can do this?

Carl (permutations) Devon (combinations)

The number of combinations is equal to the number of permutations divided by or !

A is an of elements in which the order of

the arrangement .

A is a of elements in which the order of

the selection matter.

Example

Five students take part in a cross country race to represent their school.

a. The winner of the race receives $50, the runner-up wins $25 and the third place runner wins $10. Is this an example of permutations or combinations? How many ways are there to award the three prizes? Write in two different notations.

b. For participating in the cross-country race, the school has been awarded three places at a running clinic. The school coach decides to select the 3 lucky students from the ones who took part in the cross country race. Is this an example of permutations or combinations? How many ways are there to select the three students?

The number of combinations is equal to the number of permutations divided by or !

General Rule

The number of combinations can be written as nCr or �𝑛𝑛𝑟𝑟� and is read as n choose r.

nCr = 𝑛𝑛!(𝑛𝑛−𝑟𝑟)!𝑟𝑟!

Examples

Three students from a class of twenty are to be chosen to go on a school trip. How many ways can they be selected?

To win the Lotto 6/49, a person must correctly select six numbers between 1 to 49. How many different selections of numbers can be made?

The grad committee decides to form a sub-committee of seven members to plan prom. There are a total of 15 people (9 people with short hair, 6 people with long hair) on the grad committee. The sub-committee must consist of exactly 3 people with long hair.

Determine the number of ways of selecting

People with short hair People with long hair The sub-committee

In how many ways can the sub-committee be selected if the principal must be included?

A standard deck of 52 cards is used to play many card games, including poker.

How many different five card poker hands are possible?

In how many of the hands will there be:

All diamonds? 4 black cards and 1 red card? 3 kings and 2 queens

3 kings? Four aces? 5 cards of the same suit?

Assignment: worksheet