**Problem C: Combinatorial Argument**Consider a set consisting of \( m + n \) elements, divided into two subsets: one containing \( m \) elements and the other containing \( n \) elements. We aim to demonstrate via a combinatorial argument that:\[\binom{m+n}{r} = \binom{m}{0} \binom{n}{r} + \binom{m}{1} \binom{n}{r-1} + \binom{m}{2} \binom{n}{r-2} + \ldots + \binom{m}{r} \binom{n}{0}\]Here, \(\binom{m+n}{r}\) represents the number of ways to choose \( r \) elements from a set of \( m+n \) elements. The expression on the right-hand side considers all possible ways to select \( r \) elements by choosing a different combination of elements from the two subsets. Each term \(\binom{m}{k} \binom{n}{r-k}\) corresponds to selecting \( k \) elements from the first subset (of \( m \) elements) and \( r-k \) elements from the second subset (of \( n \) elements). This summation accounts for all possible distributions of the choices across the two subsets.

m+n elements as composed of two parts.

Answered: C. Think of a set with m + n elements…

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Problem 1P

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C. Think of a set with m + n elements as composed of two parts, one with m elements and the other withn elements. Give a combinatorial argument to show that m+n' ("") = (") (") + (")(,".) + (") (,"2) + ... + (") (") m