PERMUTATION & COMBINATION

Warm up:

1.What is the probability of getting an ace and a queen assuming replacement does occur?2.What is the probability of getting a club and a spade assuming replacement does not occur?3.Suppose 6 people want to sit in 6 chairs in a row at a movie theater. In how many different ways can these 6 people be seated in the 6 available chairs?

CHECK HOMEWORK:

Given Jack, Ace, King, and Queen pick three:a) List all possible outcomes (order matter)

b) Now if order doesn’t matter how many possible outcomes are there?

DO YOU SEE A PATTERN IN a) and b) from yesterday and today?

AKJ JQK QJA AKQ

AJK JKQ QAJ AQK

KJA KQJ JAQ KQA

KAJ QKJ AJQ QKA

JKA KJQ JQA KAQ

JAK QJK AQJ QAK

IDEA OF THE DAY

How do we differentiate when to use permutation or combination rules?

INVESTIGATION:HOW MANY CHOICES DO I HAVE?

Given a King, a Jack, a Queen, and an Ace

Task 1: Students are to pick two cards at a

time order does mater. How many possible ways can a student pick 2 cards? Write down your sample space

Assume that order does not matter this time look at your sample space that you’ve just collected. How many possible ways can the student pick two of the same card

INVESTIGATION:HOW MANY CHOICES DO I HAVE?

Given a King, a Jack, a Queen, and an Ace

Task 2:o Now let’s pick three cards from

the four that are given, assuming order does matter. How many possible ways can a student pick three cards? Write down your sample space

o Assuming order doesn’t matter this time. How many ways can a student pick three cards?

GROUP DISCUSSION:

What have you notice about your sample space on the matter of whether or not order matter?

What conclusion can you make?

AMBASSADOR GAME:

Put your name down on a sheet of paper and alphabetize it

The person who are first and last in the list will go to a different group

Then discuss what you’ve discover to see if you all are in agreement with your conclusion

REVIEW: DIFFERENT METHODS

There are 3 methods for calculating the number of possible outcomes for a sequence of event.

1.Counting Rules2.Permutation Rules3.Combination Rule

REVIEW: COUNTING PRINCIPLE:

Suppose that two events occur in order. If the first can occur in “m” ways and the second in “n” ways (after the first has occurred), then the two events can occur in order in m x n ways.

Order is extremely important for the counting principle.

In simple words, multiply the number of possibilities for each individual event

REVIEW FACTORIAL:

PermutationsPermutations

A A permutationpermutation of a set of of a set of distinct objects is an distinct objects is an orderingordering of of these objects.these objects.

Used for an arrangement of Used for an arrangement of object that are in specific orderobject that are in specific order

The number of permutations of n The number of permutations of n objects is n!objects is n!

ExampleExample

For example, here are some For example, here are some permutations of the word “THEIS”:permutations of the word “THEIS”:

SIEHTSIEHT HEISTHEIST HTSEIHTSEI TISEHTISEH

etc.etc.

Example:Example:

A club has nine members. In how A club has nine members. In how many ways can a president, vice many ways can a president, vice president, and secretary be president, and secretary be chosen from the members of this chosen from the members of this club?club?

Permutations of Permutations of nn objects taken objects taken rr at a at a timetime

The number of permutations of The number of permutations of nn objects taken objects taken rr at a time is: at a time is:

On your calculator, permutations are On your calculator, permutations are written as nPr and are found under written as nPr and are found under MATHMATHPRB.PRB.

)!(

!),(

rn

nrnP

EXAMPLES:

How many different ways can a chairperson and an assistant chairperson be selected for a research project if there are seven scientists available?

EXAMPLES:

How many permutations can be formed from the word “Justice” using only 5 letters?

EXAMPLE:

From a group of 9 different books, 4 books are to be selected and arranged on a shelf. How many arrangements are possible?

CombinationsCombinations

Used for an arrangement of objects with Used for an arrangement of objects with No specific orderNo specific order

Order doesn’t matterOrder doesn’t matter

CombinationsCombinations

The formula for computing a combination The formula for computing a combination of of nn objects taking objects taking rr at a time is:at a time is:

On your calculator, you can use nCr On your calculator, you can use nCr which is located under MATHwhich is located under MATHPRBPRB

)!(!

!),(

rnr

nrnC

EXAMPLE:

How many combinations of the letter A,B,E,H,L and P are there if they put in groups of 2s?

EXAMPLE:

At the hamburger Hut you can order hamburgers with cheese, onions, pickles, relish, mustard, lettuce or tomato. How many different combinations of the “extras” can you order, choosing any 3 of them?

EXAMPLE:

In a club there were 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. Hoe many different possibilities are there?

REVISITING THE EQ:

When do I use the counting principle?

HOMEWORK 13.1