Counting techniuqes
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Counting Techniuqes. Counting Techniques. Sample Spaces List all outcomes and count Organized list Tree diagrams Filling in blanks. Create Sample Space For Flipping a coin and rolling a die. Tree diagram H T 1 2 3 4 5 6 1 2 3 4 5 6

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Counting Techniuqes

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Counting techniuqes

Counting Techniuqes


Counting techniques

Counting Techniques

Sample Spaces

List all outcomes and count

Organized list

Tree diagrams

Filling in blanks


Create sample space for flipping a coin and rolling a die

Create Sample Space For Flipping a coin and rolling a die

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


Flipping coins

Flipping Coins

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 ___ ____


Questions to ask

Questions to Ask

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?


Flipping 3 coins

Flipping 3 Coins

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

____ ____ ____


Flipping coins1

Flipping Coins

  • 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


Number of outfits

Number of Outfits

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

Tree diagram or

___ ___ ___

3 * 2 * 4


Rolling die

Rolling Die

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?


How many ways can 6 people line up in a row

How Many Ways Can 6 People Line Up in a Row?

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


Question

Question

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


6 people in a row

6 People in a Row

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


Extension

Extension

  • 40 people into 3 seats

  • ___ ___ ___

  • Another way to think of this is

  • 40*39*38*37*36*……1

  • 37*36*35……1

  • 40!/37!


What about this one

What about this one?

  • 40 people into 15 seats

  • 40*39*…..26

  • or

  • 40!/(40-15)!

  • 40!/25!


Arranging letters in words

Arranging Letters in Words

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?


Arranging letters in words1

Arranging Letters in Words

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!


Arranging letters in words2

Arranging Letters in Words

What about MISSISSIPPI?

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


License plates

License Plates

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


Class officers

Class Officers

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!


Class officers1

Class Officers

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!


Permutations

Permutations

For problems where order is important.

12 people in 5 chairs

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


Teams or committees

Teams or Committees

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)


Teams or committees1

Teams or Committees

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!)


Teams or committees2

Teams or Committees

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!)


Combinations

Combinations

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


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