   Chapter 9.2, Problem 44ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Prove Theorem 9.2.1 by mathematical induction.

To determine

To prove the below theorem using mathematical induction.

“If an operation consists of k steps and the first step can be performed in n1 ways, the second step can be performed in n2 ways [regardless of how the first step was performed],

The kth step can be performed in nk ways [regardless of how the preceding steps were performed],

then the entire operation can be performed in n1n2...........nk ways.”

Explanation

Given information:

The entire operation can be performed in n1n2.......nk ways

Concept used:

P(n.r)=n!nr!

Proof:

To prove theorem by mathematical induction using 2 steps as follows:

The first step can be performed in n1 ways. The second step can be performed in n2 ways.

This implies that the entire operation can be performed in n2+n2+.........n1 times it is

=n1×n2

Therefore, the result ¡f proved for n=2.

Let us assume that the result is true for n=k.

i.e., if there are k steps, then the operation can be performed in n1n1

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