1. Main Menu Main Menu (Click on the topics below) Algebra of Combinations Pascal’s formula More Formulas Click on the picture Sanjay Jain, Lecturer, School of Computing

2. Algebra of Combinations Sanjay Jain, Lecturer, School of Computing

3. Algebra of Combinations 1. Simplification 2. Proving some equalities. 3. Different ways of seeing the same problem. For r > n For 1  r  n For 0  n

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5. Theorem: Suppose 0  r  n Method 2: Choosing r objects from a set of n objects is same as leaving out n-r objects. Thus, Method 1:

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7. Theorem: Pascal’s Formula Job: Choose r objects from a set of n+1 objects. Calculate in two ways: A) B) (i) Choose the first object, and r-1 of the remaining n objects OR (ii) Do not choose the first object, and choose r of the remaining n objects Answer should be sum of the ways in which (i) and (ii) can be done. Suppose 1  r  n

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9. Pascal’s Formula Continued We can use Pascal’s Formula to derive new formulas as follows: Suppose 2  r  n

10. Pascal’s Formula Continued Choosing r out of n+2 objects can be done as follows: (i) Choose the first two objects, and r-2 of the remaining n objects OR (ii) Choose one of the first two objects and choose r-1 of the remaining n objects OR (iii) Choose none of the first two objects and choose r of the remaining n objects. Number of ways of choosing r out of n+2 objects should be sum of the ways in which (i), (ii) and (iii) can be done.

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12. Generalized Pascal’s Formula Generalizing we get:

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14. More Formulas You can prove above by substituting the formulas for combinations. Problem: Choose a committee of size r+1, with a leader, from a group of size n. A) Choose a committee of size r+1, and then choose a leader among them. B) Choose a committee of size r, and then choose a leader among the remaining n-r people. Answer should be same from both the methods. Suppose 0  r  n

15. More Formulas A) Choose a committee of size r+1, and then choose a leader among them. Choosing a committee of size r+1: Choosing a leader: (r+1) ways. Choosing the committee with the leader:

16. More Formulas B) Choose a committee of size r, and then choose a leader among the remaining n-r people. Choosing a leader: (n - r) ways. Choosing a committee of size r (without leader): Choosing the committee with the leader:

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