36
Counting Techniuqes
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 • H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6• Filling in Blanks• Coin Die• ____ ____• 2 * 6
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
• 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
• 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 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
• 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
• 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?
• 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
• How is arranging 4 people in a row different than flipping a coin 4 times?
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
• 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?
• 40 people into 15 seats
• 40*39*…..26
• or
• 40!/(40-15)!
• 40!/25!
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 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 Words
• What about MISSISSIPPI?
• 11!/(4! * 4! * 2!)=
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
• 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 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
• For problems where order is important.
• 12 people in 5 chairs
• At a much later time after lots of problems, introduce permutations.
•
!7
!12
)!512(
!12512
P
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 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 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
• At a much later time after lots of problems, introduce combinations.
•
!5!7
!12
!5)!512(
!12512
C
40!
25!
40! 40
25! 25
40! 40 8
25! 25 5
40! 40 81.6
25! 25 5
11!
4!4!2!
11! 11
4!4!2! 4!4!2
11! 11 11
4!4!2! 4!4!2 4 4 2!!
11! 11 11
4!4!2! 4!4!2 4 4 2!!
11
32!!
11! 11 11
4!4!2! 4!4!2 4 4 2!!
11 11
32!! 32
ii
11! 11 11
4!4!2! 4!4!2 4 4 2!!
11 11 11( 1)
32!! 32 32
ii
11! 11 11
4!4!2! 4!4!2 4 4 2!!
11 11 11( 1) 11
32!! 32 32 32
ii
