La Derive Newton's iden tity() in termsb.) Use part ca) toitsofexpressDIeredecessor

To prove nkkr=nrn-rk-r , n≥k≥0 L.H.S…

Answered: 1)a Derive Newton's iden tity b.) Use…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

La Derive Newton's iden tity
() in terms
b.) Use part ca) to
its
of
express
DIeredecessor

Expert Solution

part a

To prove $(\begin{matrix} n \\ k \end{matrix}) (\begin{matrix} k \\ r \end{matrix}) = (\begin{matrix} n \\ r \end{matrix}) (\begin{matrix} n - r \\ k - r \end{matrix}), n \geq k \geq 0$

L.H.S

$\begin{array}{rcl} (\begin{matrix} n \\ k \end{matrix}) (\begin{matrix} k \\ r \end{matrix}) & = & \frac{n!}{k! (n - k)!} \frac{k!}{r! (k - r)!} \\ = & \frac{n!}{r! (n - k)! (k - r)!} \\ = & \frac{n!}{r! (n - r)!} \frac{(n - r)!}{(n - k)! (k - r)!} \\ = & \frac{n!}{r! (n - r)!} \frac{(n - r)!}{(k - r)! [(n - r) - (k - r)]!} \\ = & (\begin{matrix} n \\ r \end{matrix}) (\begin{matrix} n - r \\ k - r \end{matrix}) \\ R . H . S \end{array}$

Hence proved.

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