   Chapter 9.7, Problem 16ES ### 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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# Think of a set with m + n elements as composed of two parts, one with m elements and the other with n elements. Give a combinatorial argument to show that ( m + n r ) ( m 0 ) ( n r ) + ( m 1 ) ( m r − 1 ) + ⋯ + ( m r ) ( n 0 ) where m and n are positive integers and r is an integer that is less than or equal to both m and n. This identity gives rise to many useful additional

To determine

Give a combinatorial argument to show that;

( m+nr)=(m0)(nr)+(n1)(m r1)++(mr)(n0)

where m and n are positive integers and r is an integer that is less than or equal to both m and n.

Explanation

Given information:

Think of a set with m+n elements as composed of two parts, one with m elements and the other with n elements.

Consider a set of elements m+n (here m,nZ+ ). Let rZ+ such that rm,n.

Number of combinations to select r number of elements from m+n=( m+nr)(1).

Let a set of m+n elements be composed of two parts, one with m elements and the other with n elements.

Combinations to select r number of elements such that i ( Z and ir ) number of elements are selected from set of m and ri number of elements are selected from set of n ;

=i=0r(mi)(n ri)

=(

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