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 ___ ___ ___ ___ ___ 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.

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.