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Combinations

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ExampleTheorem

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CombinationsCombinations

Sanjay Jain, Lecturer, Sanjay Jain, Lecturer,

School of ComputingSchool of Computing

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).

CombinationsCombinations

Combinations ---> unordered selectionPermutation ---> ordered selection

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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

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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

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

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

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

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

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

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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?

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

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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?

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

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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.

ExampleExample

Case 1: Tom is in the team.

10

298

Need to select 10 out the remaining 298 students.

ExampleExample

Case 2: Sam is in the team.

10

298

Need to select 10 out the remaining 298 students.

ExampleExample

Case 3: Both Tom and Sam are not in the team.

11

298

Need to select 11 out the remaining 298 students.

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:

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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.

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:

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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?

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

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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

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