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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

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Algebra of Combinations Sanjay Jain, Lecturer, School of Computing

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

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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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Generalized Pascal’s Formula Generalizing we get:

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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

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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:

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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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More Formulas

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Algebraic Proof Addition:If a = b, then a + c = b + c. Subtraction:If a = b, then a - c = b - c. Multiplication: If a = b, then ca = cb. Division: If a.

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