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Induction and Basic Counting

# Induction and Basic Counting - Math 100 Induction and...

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Math 100 Induction and Counting Martin H. Weissman 1. Basic Counting 1.1. The Handshake Problem. Suppose that forty people enter a room. Each person shakes each other person’s hand, exactly once. How many handshakes took place? What if there were a hundred people? Can you find a formula, for the number of handshakes, given n people? 1.2. Choice-Sequences. Suppose that we have a finite sequence of “choices” to make, and we wish to find out how many “total choices” there are. We illustrate this with the example of “how to dress”: I have four pairs of jeans in my drawer. I have to choose one to wear. I have seven shirts in my closet. I have to choose one to wear. I have two pairs of shoes. I have to choose one (pair) to wear. How many ways are there of getting dressed? We use the following two basic principles : If n is a positive integer, then the number of ways to choose 1 object from a collection of n objects is equal to n . If we make a final sequence of (independent) choices, then the total number of outcomes is equal to the product of the numbers of choices at each step in the sequence.

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