# Note that a product x 1 x 2 x 3 may be parenthesized in two different ways: ( x 1 x 2 ) x 3 and x 1 ( x 2 x 3 ) . Similarly, there are several different ways to parenthesize x 1 x 2 x 3 x 4 . Two such ways are ( x 1 x 2 ) ( x 3 x 4 ) and x 1 ( ( x 2 x 3 ) x 4 ) . Two such ways are ( x 1 x 2 ) ( x 3 x 4 ) and x 1 ( ( x 2 x 3 ) x 4 ) . Let P n be the number of different ways to parenthesize the product x 1 x 2 ... x 4 . Show that if P 1 = 1 , then P n = ∑ k = 1 n = 1 P k P n − k for every integer n ≥ 2. (It turns out that the sequence P 1 , P 2 , P 3 , ... is the same as the sequence of Catalan numbers: P n = C n − 1 for every integer n ≥ 1. See example 5.6.4.)

### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193

### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193

#### Solutions

Chapter
Section
Chapter 9.3, Problem 44ES
Textbook Problem

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