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Click on the picture. Main Menu (Click on the topics below). Combinations Example Example Example Example Example Example Theorem. Combinations. Sanjay Jain, Lecturer, School of Computing. Let n, r  0, be such that r n.

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  1. Click on the picture Main Menu (Click on the topics below) Combinations Example Example Example Example Example Example Theorem

  2. Combinations Sanjay Jain, Lecturer, School of Computing

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

  4. Combinations ---> unordered selection Permutation ---> ordered selection Combinations

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  6. 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 Example

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  8. In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge. Example 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

  9. In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge. T1: can be done in ways. Example T1: give 13 cards to N

  10. In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge. Example T2: give 13 of the remaining cards to E T2: can be done in ways.

  11. In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge. T3: can be done in ways. Example T3: give 13 of the remaining cards to S

  12. In how many ways 52 cards can be distributed to N, S, E, W in a game of bridge. T4: can be done in ways. Example T4: give 13 of the remaining cards to W

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

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  15. 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

  16. T1: Choose curriculum committee T2: Choose exam committee ways The selection of both comm can be done in * Example T1: ways T2: ways

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  18. From 300 students I need to select a president, secretary and 3 ordinary members of Executive committee. How many ways can this be done? Example

  19. 1st Method: T1: president---300 T2: secretary---299 T3: 3 ordinary members---298C3 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

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  21. 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

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

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

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

  25. 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 Example Thus total number of possible ways to select the team is:

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  27. 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. Example

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

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  30. 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

  31. 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 Example

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  33. Proof: Theorem Choose k out of n elements P(n,k) Choose k out of n elements in order a) Choose k out of n elements. b) Put order Thus:

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