(a) Use a counting argument to prove that- () )()(+++(b) Generalize this as far as you can.

(a) use a counting argument to prove that' 85=5233+5332+5431+5530 it is known that nr=n!r!n-r! left…

Answered: (a) Use a counting argument to prove…

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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Related questions

Question

(a) Use a counting argument to prove that
- () )
()(
+
+
+
(b) Generalize this as far as you can.

Expert Solution

Step 1

(a) use a counting argument to prove that'

$(\begin{matrix} 8 \\ 5 \end{matrix}) = (\begin{matrix} 5 \\ 2 \end{matrix}) (\begin{matrix} 3 \\ 3 \end{matrix}) + (\begin{matrix} 5 \\ 3 \end{matrix}) (\begin{matrix} 3 \\ 2 \end{matrix}) + (\begin{matrix} 5 \\ 4 \end{matrix}) (\begin{matrix} 3 \\ 1 \end{matrix}) + (\begin{matrix} 5 \\ 5 \end{matrix}) (\begin{matrix} 3 \\ 0 \end{matrix})$

it is known that

$(\begin{matrix} n \\ r \end{matrix}) = \frac{n!}{r! (n - r)!}$

left hand side

$(\begin{matrix} 8 \\ 5 \end{matrix}) = \frac{8!}{5! (8 - 5)!} = \frac{8!}{5! \times 3!} = \frac{8 \times 7 \times 6 \times 5!}{5! \times 3 \times 2 \times 1} = 56$