Main Menu (Click on the topics below)
-
Upload
jorden-guzman -
Category
Documents
-
view
27 -
download
0
description
Transcript of Main Menu (Click on the topics below)
![Page 1: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/1.jpg)
Main Menu Main Menu (Click on the topics below)
Combinations
Example
Example
Example
Example
Example
ExampleTheorem
Click on the picture
![Page 2: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/2.jpg)
CombinationsCombinations
Sanjay Jain, Lecturer, Sanjay Jain, Lecturer,
School of ComputingSchool of Computing
![Page 3: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/3.jpg)
CombinationsCombinations
Let n, r 0, be such that r n. Suppose A is a set of n elements. An r-combination
of A, is a subset of A of size r.
r
n Pronounced: n choose rDenotes the number of different r-combinations of a set of size n.
Some other notations commonly used are nCr, and C(n,r).
![Page 4: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/4.jpg)
CombinationsCombinations
Combinations ---> unordered selectionPermutation ---> ordered selection
![Page 5: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/5.jpg)
END OF SEGMENT
![Page 6: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/6.jpg)
ExampleExample
There are 7 questions in an exam.You need to select 5 questions to answer.How many ways can you select the questions to
answer?(order does not matter)
5
7 Here 7 is the number of questions and 5 is the number of questions selected
![Page 7: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/7.jpg)
END OF SEGMENT
![Page 8: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/8.jpg)
ExampleExample
In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge.
T1: give 13 cards to N
T2: give 13 of the remaining cards to E
T3: give 13 of the remaining cards to S
T4: give 13 of the remaining cards to W
![Page 9: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/9.jpg)
ExampleExample
In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge.
T1: give 13 cards to N
T1: can be done in ways.
13
52
![Page 10: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/10.jpg)
ExampleExample
In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge.
T2: give 13 of the remaining cards to E
T2: can be done in ways.
13
39
![Page 11: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/11.jpg)
ExampleExample
In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge.
T3: give 13 of the remaining cards to S
T3: can be done in ways.
13
26
![Page 12: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/12.jpg)
ExampleExample
In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge.
T4: give 13 of the remaining cards to W
T4: can be done in ways.
13
13
![Page 13: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/13.jpg)
ExampleExample
In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge.
T1: give 13 cards to N
T2: give 13 of the remaining cards to E
T3: give 13 of the remaining cards to S
T4: give 13 of the remaining cards to W
Thus using the multiplication rule total number of ways in which cards can be distributed is
13
13*
13
26*
13
39*
13
52
![Page 14: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/14.jpg)
END OF SEGMENT
![Page 15: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/15.jpg)
ExampleExample
70 faculty members.Need to choose two committees:
A) Curriculum committee of size 4
B) Exam committee of size 3
How many ways can this be done if the committees are to be disjoint?
![Page 16: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/16.jpg)
ExampleExampleT1: Choose curriculum committee
T2: Choose exam committee
T1:
4
70 ways
T2:
3
66ways
The selection of both comm can be done in
4
70
3
66* ways
![Page 17: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/17.jpg)
END OF SEGMENT
![Page 18: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/18.jpg)
ExampleExample
From 300 students I need to select a president, secretary and 3 ordinary members of Executive committee.
How many ways can this be done?
![Page 19: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/19.jpg)
1st Method:
T1: president---300
T2: secretary---299
T3: 3 ordinary members---298C3
3
298*299*300
2nd Method:
T1: 5 members of the committee --- 300C5
T2: choose president among the members of the committee --- 5
T3: choose secretary among the members of the committee --- 4
4*5*5
300
![Page 20: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/20.jpg)
END OF SEGMENT
![Page 21: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/21.jpg)
ExampleExample
From 300 students I need to select a football team of 11 players.
Tom and Sam refuse to be in the team together.How many ways can the team be selected?
Case 1: Tom is in the team.Case 2: Sam is in the team.Case 3: Neither Tom nor Sam is in the team.
![Page 22: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/22.jpg)
ExampleExample
Case 1: Tom is in the team.
10
298
Need to select 10 out the remaining 298 students.
![Page 23: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/23.jpg)
ExampleExample
Case 2: Sam is in the team.
10
298
Need to select 10 out the remaining 298 students.
![Page 24: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/24.jpg)
ExampleExample
Case 3: Both Tom and Sam are not in the team.
11
298
Need to select 11 out the remaining 298 students.
![Page 25: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/25.jpg)
ExampleExample
From 300 students I need to select a football team of 11 players.
Tom and Sam refuse to be in the team together.How many ways can the team be selected?
Case 1: Tom is in the team. --- 298C10
Case 2: Sam is in the team. --- 298C10
Case 3: Neither Tom nor Sam is in the team. --- 298C11
11
298
10
298
10
298
Thus total number of possible ways to select the team is:
![Page 26: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/26.jpg)
END OF SEGMENT
![Page 27: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/27.jpg)
ExampleExample
There are 6 boys and 5 girls. In how many ways can one form an executive committee of size 4 such that there is at least one member of each sex?
Wrong method:
T1: select one boy. --- 6 ways
T2: select one girl. --- 5 ways
T3: select 2 others. --- 9C2 ways
6*5* 9C2 ways to select the committee.
![Page 28: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/28.jpg)
ExampleExample
There are 6 boys and 5 girls. In how many ways can one form an executive committee of size 4 such that there is at least one member of each sex?
Wrong method:
B1G1G2, G3
B1G2G1, G3
B1G3G1, G2
Selection of B1, G1, G2, G3 is counted as:
![Page 29: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/29.jpg)
END OF SEGMENT
![Page 30: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/30.jpg)
ExampleExample
There are 6 boys and 5 girls. In how many ways can one form an executive committee of size 4 such that there is at least one member of each sex?
![Page 31: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/31.jpg)
ExampleExampleCorrect method: A: Choose 4 members of the committee (without
restrictions) B: Choose 4 members of the committee without any
boys. C: Choose 4 members of the committee without any
girls.D: Choose 4 members of the committee with at least
one boy and at least one girl.
D=A-B-C
4
11
4
5
4
6
4
6
4
5
4
11
![Page 32: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/32.jpg)
END OF SEGMENT
![Page 33: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/33.jpg)
TheoremTheorem
Proof:
)!(!
!
!
),(
knk
n
k
knP
k
n
Choose k out of n elements
Choose k out of n elements in order
a) Choose k out of n elements.b) Put order
P(n,k)
k
n
Thus:
!*),( kk
nknP
k
n
!k
![Page 34: Main Menu (Click on the topics below)](https://reader030.fdocuments.in/reader030/viewer/2022032605/56812b94550346895d8fb6e6/html5/thumbnails/34.jpg)
END OF SEGMENT