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Combinations Sanjay Jain, Lecturer, School of Computing

Combinations 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. Pronounced: n choose r Denotes the number of different r-combinations of a set of size n. Some other notations commonly used are n C r, and C(n,r).

Combinations Combinations ---> unordered selection Permutation ---> ordered selection

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Example 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) Here 7 is the number of questions and 5 is the number of questions selected

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Example In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge. T 1 : give 13 cards to N T 2 : give 13 of the remaining cards to E T 3 : give 13 of the remaining cards to S T 4 : give 13 of the remaining cards to W

Example In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge. T 1 : give 13 cards to N T 1 : can be done in ways.

Example In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge. T 2 : give 13 of the remaining cards to E T 2 : can be done in ways.

Example In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge. T 3 : give 13 of the remaining cards to S T 3 : can be done in ways.

Example In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge. T 4 : give 13 of the remaining cards to W T 4 : can be done in ways.

Example In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge. T 1 : give 13 cards to N T 2 : give 13 of the remaining cards to E T 3 : give 13 of the remaining cards to S T 4 : give 13 of the remaining cards to W Thus using the multiplication rule total number of ways in which cards can be distributed is

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Example 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?

Example T 1 : Choose curriculum committee T 2 : Choose exam committee T1:T1: ways T2:T2: The selection of both comm can be done in * ways

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Example 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: T 1 : president T 2 : secretary T 3 : 3 ordinary members C 3 2nd Method: T 1 : 5 members of the committee C 5 T 2 : choose president among the members of the committee T 3 : choose secretary among the members of the committee --- 4

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Example 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.

Example Case 1: Tom is in the team. Need to select 10 out the remaining 298 students.

Example Case 2: Sam is in the team. Need to select 10 out the remaining 298 students.

Example Case 3: Both Tom and Sam are not in the team. Need to select 11 out the remaining 298 students.

Example 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 C 10 Case 2: Sam is in the team C 10 Case 3: Neither Tom nor Sam is in the team C 11 Thus total number of possible ways to select the team is:

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Example 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: T 1 : select one boy ways T 2 : select one girl ways T 3 : select 2 others C 2 ways 6*5* 9 C 2 ways to select the committee.

Example 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: B1 G1 G2, G3 B1 G2 G1, G3 B1 G3 G1, G2 Selection of B1, G1, G2, G3 is counted as:

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Example 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?

Example Correct 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

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Theorem Proof: 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) Thus:

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