Formula 1: Example: Total: Example: (75 ) (76) N (N +1) = (20 ) (21 ) =1050

Transcription

1 Formula 1: S= N Example: Total: N (N +1) S= (75 ) (76) =850 Example: ( ) 5 (0 ) (1 ) =

2 1++3=6, so add 1 through 5 and subtract 6 from answer. 1+3 (5 ) (6 ) = = (5 ) (6 ) 6= =? S= =? S=7( ) S= 7 (60) (61) =1810 Final answser: =1803

3 We want: T = =? We know how to get: S= S=T T =S 30 S=5( ) S= 5 (0) (1) so T =S 30= =100

4 What is N? S= N=340 S=340, and N (N +1) S= So we must have N ( N +1 ) =340 Multiply by : get Logic says: N <N (N +1 )< (N +1 ) So N <6480<( N +1 ) Take the square root: N < 6480<( N +1 ) N (N +1)=6480

5 N <80.5<(N +1) Using logic, N=80 Check it!!! Want N S= N= N ( N +1 ) (80) ( 81) = =340, as expected. Given S, find the stopping point N by N < S We know S= N (N +1), so we must have S=N (N +1) If we take N < S, find the square root and Drop the decimal to get N.

6 What is N? S= N=5050 From the formula, we know N ( N +1) =5050 Multiply both sides by : get N (N +1)=10100 N (N +1) is a little more than N N, so the number N is a bit less than the square root of (as an example, Notice 6*7 is a bit more than 6*6) N <N (N +1 )=10100 N < (equals about) So N=100. Remember to check!!!

7 N N ( N +1) Variations: =7( ) 7 (0 ) (1 ) = ( ) ( ) 50 (51 ) 4 (5) = 50 (51 ) 4 (5 ) = N =5151 what is N?

8 5151= N ( N +1 ) 1030=N ( N +1) N N So N=101 (always check!!) 101 (10 ) =5151 New stuff: In an election with 5 candidates, A, B, C, D, E, how many headto-head comparisons are there? (e.g. A vs B, A vs C, etc)

9 A vs B A vs C B vs C A vs D B vs D C vs D A vs E B vs E C vs E D vs E = Number of head-to-head comparisons for 5 candidates Notice the sum for 5 candidates starts with 4. (one less than the number of candidates.) Number of head-to-head comparisons for 5 candidates: = 4 (5) =10 Number of head-to-head comparisons for N candidates?

10 ( N 1) ( N 1+1) ( N 1)= ( N 1) N 5 teams, (A, B, C, D, E), Every team plays every other team and themselves (scrimage). Count number of games scheduled. A vs A A vs B B vs B A vs C B vs C C vs C A vs D B vs D C vs D D vs D A vs E B vs E C vs E D vs E E vs E

11 Here we have 5 options (candidates), since we included playing against yourself, our N is still = 5 (6) Total number of match-ups (including against yourself) with N players is: N ( N +1 )

12 Total match-ups if you cannot play against yourself with N players is: N ( N 1 ) Head-to-head without self-pairing = pairwise comparison An election requires 36 pairwise comparisons. How many candidates were there?

13 We know the total will be (N 1), where N is the number of candidates ( N 1) ( N 1) ( N 1+1 ) = ( N 1) N =36 So we get 7=( N 1 ) N N So N=9 Check: 9 candidates, pairwise comparison: = 8 (9 ) =36

14 Counting number of ways to do a task. 1) Break the problem up into non-overlapping subtasks Example: Count ways to make a license plate of the form ABC13 (three letters followed by three numbers) Task 1: Choose the first letter Task : Choose the nd letter Task 3: Choose the 3 rd letter Task 4: choose 1 st number Task 5: choose nd number Task 6: choose 3 rd number ) Count the number of ways to do each subtask.

15 In our example, each of the first 3 subtasks have 6 ways. The last three subtasks have 10 ways each (digits 0 through 9). 3) If the original task is completed by doing each subtask, (must do ALL subtasks to complete), then the total ways to complete the original task is the product of the subtasks. (multiply the subtasks) The answer to our example is 6*6*6*10*10*10 i.e =17,576,000

16 How many ways to make the license plate of the form ABC13 if each character must be distinct (no repeated letters or numbers)? 1 st task: 6 ways (pick any letter) nd task: 5 ways (pick any letter not the same as the 1 st ) 3 rd task: 4 ways 4 th task: 10 (pick a number 0-9) 5 th task: 9 (pick a different number) 6 th task: 8 Total: = 11,3,000

17 (continuing steps to count ways to complete task) 4) If the task is completed by doing exactly one of the subtasks, then the number of ways to do the original task is the sum of the subtasks (add the subtasks together)

18 Example: A password character must be a digit, an upper or lower case letter, or one of the special characters +, *,!, or?. a) How many password characters are possible? b) How many 4 character passwords are possible? c) How many password characters are possible? Task= choose a character Subtask 1: choose a digit OR Subtask : choose a lower case letter OR Subtask 3: choose a capital letter

19 OR Subtask 4: choose a special character -Only need to complete ONE subtask to complete original task, so we need to add the subtasks. Total ways to pick a character: = 66 How many 4 character passwords are possible? Recall: 66 possible characters Subtask 1: pick 1 st character AND Subtask : pick nd character AND Subtask 3: pick 3 rd character

20 AND Subtask 4: pick 4 th character Since we need to complete ALL subtasks to complete the original task, we need to mulitply the subtasks. Total ways to make a password: 664 = 18,974,736 Comments: OR = add (do task 1 OR task ) AND = multiply (do task 1 AND task )

21 How many legs does a horse have? Answer: 8 Proof: two on the right, two on the left, two in the front, two in the back, so +++=8 (example of counting mistake) These subtasks overlapped. Make sure your subtasks do not overlap! (do not count the same thing more than once) Notation: 0!=1, and 1!=1, 3!=1 3 N!=1 3 4 (N 1) N How many ways to arrange 5 books on a shelf:

22 5!= How many ways to arrange ABCDE so that it is not alphabetical order? Task 1: pick 1 st letter Task : nd letter (5 choices) (4 choices)

23 Task 5: pick last letter (1 choice) TOTAL ways to arrange ABCDE: 5!= =10 But we don t want to count the arrangement that is in alphabetical order (only 1 arrangement in order) The answer to the original question is 10 1=119 Factorial example: (7 4 )! 3! 3 1 6

24 Soups: Veg or Pea Salad: Garden, caesar, fruit, pasta Soup and salad: *4= 8 possible pairings Veg+garden, veg+caesar, veg+fruit, veg+pasta OR Pea+garden, pea+caesar, pea+fruit, pea+pasta = 8 possible pairs N = N ( N +1 ) = 10 ( )=

25 10 (100 ) (101) =5(100)(101) N =5050 what is N? N ( N +1) =5050 Multiply through by N (N +1)=10100

26 Now N (N +1) is LIKE N (its just a little more than N So N <10100 N < Hence N=100 (always check!)