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Counting Techniuqes

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Sample Spaces

List all outcomes and count

Organized list

Tree diagrams

Filling in blanks

Tree diagram

H T

1 2 3 4 5 6 1 2 3 4 5 6

H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6

Filling in Blanks

Coin Die

____ ____

2 * 6

One Coin H T

Two Coins HH TH TT

Perform experiments

Tree Diagram

H T

H T H T

HH, HT, TH, TT

Filling Blanks ___ ____

What did your group do to solve this?

Can anyone find a counter example?

How do you know this answer works?

Is this consistent with what we found earlier?

Does anyone have another way to explain?

How is this similar or different from previous problems?

Tree Diagram

H T

H T H T

H T H T H T H T

HHH, HHT, HTH, HTT, THH, THT, TTH, TTT

Organized list

Fill in the blanks

____ ____ ____

- Create Sample space for 4 coins
- Number in sample space for 5 coins
- Number in sample space for 10 coins
- Extension activity
- Pascal’s Triangle

3 shirts, 2 pants and 4 shoes and they all “match.” How many different outfits could you wear?

Tree diagram or

___ ___ ___

3 * 2 * 4

Sample space for rolling 1 dice

Sample space for rolling 2 die

# in sample space for rolling 3 die?

Rolling 3 die what is the probability that the sum of the die is more than 4?

Let’s begin with a simpler problem and find a pattern.

2 people—A and B

AB or BA

3 people—A, B, and C

ABC, ACB, BAC, BCA, CAB, CBA

Use the sample space above to find the number for 4 people—A, B, C, and D

How is arranging 4 people in a row different than flipping a coin 4 times?

5 people with 5 positions

___ ___ ___ ___ ___

5 * 4 * 3 * 2 * 1

6 people with 6 positions

6! = 6*5*4*3*2*1

What if we had 6 people but only 4 seats?

___ ___ ___ ___

6 5 4 3

- 40 people into 3 seats
- ___ ___ ___
- Another way to think of this is
- 40*39*38*37*36*……1
- 37*36*35……1
- 40!/37!

- 40 people into 15 seats
- 40*39*…..26
- or
- 40!/(40-15)!
- 40!/25!

How many “words” can be created using the letters in MATH?

___ ___ ___ ___

4 3 2 1

What about the word BOOK? Is it the same?

In sample space is BKOO and BKOO

How many of the 24 “words” are duplicates?

How many “words” for ALGEBRA?

7!/2

What about a word like BOOKO?

5!/3

One “word” is BKOOO. How many ways can three “O” be arranged?

__ __ __ = 3 * 2 * 1 = 3! = 6

5!/3!

What about MISSISSIPPI?

11!/(4! * 4! * 2!)=

5 spaces

___ ___ ___ ___ ___

Use digits 0 – 9

10 ^5

What if numbers can not be repeated?

What if we use letters in the first two spots?

26 * 26 * 10 *10 * 10

12 names and president, vice-president

12 * 11

As we discovered before

12 * 11 * 10 * …….*1

10 * 9 * 8 * ……* 1

Or 12!/(12-2)! Or 12!/10!

I have a president, vice-president, secretary and treasure to elect. Names will be placed in a hat and names drawn out.

How many different sets of officers can created if there are 10 names in the hat?

___ ___ ___ ___

10 * 9 * 8 * 7

Or 10!/(10-4)! Or 10!/6!

For problems where order is important.

12 people in 5 chairs

At a much later time after lots of problems, introduce permutations.

I have 6 people and I want to form a team of 3 people. How many different teams?

___ ___ ___

6 * 5 * 4

One team is A B C

Is B A C a different team?

How many ways can A B C be arranged?

3*2*1 = 6

(6 * 5 * 4)/(3 * 2 * 1)

How many ways can a 5 person team be formed from 12 people?

If order were important

(12*11*10*9*8) or 12!/(12-5)! Or 12!/7!

Because order is not important we need to divide the above answer by 5!

(12*11*10*9*8)/5!

12!/7!/5!

12!/(7! * 5!)

20 people and 5 person team

How many teams with duplicates

20*19*18*17*16 or 20!/(20-5)! Or 20!/15!

Because order is not important we need to divide the above answer by 5!

20!/15!/5!

20!/(15! * 5!)

At a much later time after lots of problems, introduce combinations.