Q. 1.9 Use Theoretical Exercise 8 to pr... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 1: Q. 1.9 (page 17)

Use Theoretical Exercise 8 to prove that
$(\begin{matrix} 2 n \\ n \end{matrix}) = {鈭�}_{k = 0}^{n} {(\begin{matrix} n \\ k \end{matrix})}^{2}$

Short Answer

Expert verified

It is proved that $(\begin{matrix} 2 n \\ n \end{matrix}) = {鈭�}_{k = 0}^{n} {(\begin{matrix} n \\ k \end{matrix})}^{2}$

Step by step solution

Step 1. State the Theoretical Exercise 8.

According to Theoretical Exercise 8,

$(\begin{matrix} n + m \\ r \end{matrix}) = (\begin{matrix} n \\ 0 \end{matrix}) (\begin{matrix} m \\ r \end{matrix}) + (\begin{matrix} n \\ 1 \end{matrix}) (\begin{matrix} m \\ r - 1 \end{matrix}) + . . . . + (\begin{matrix} n \\ r \end{matrix}) (\begin{matrix} m \\ 0 \end{matrix})$ ...................... (1)

Step 2. Prove the given equation.

By substituting $m = n$ and $r = n$ in equation (1), we get

role="math" localid="1647934551213" $(\begin{matrix} n + n \\ n \end{matrix}) = (\begin{matrix} n \\ 0 \end{matrix}) (\begin{matrix} n \\ n \end{matrix}) + (\begin{matrix} n \\ 1 \end{matrix}) (\begin{matrix} n \\ n - 1 \end{matrix}) + . . . . + (\begin{matrix} n \\ n \end{matrix}) (\begin{matrix} n \\ 0 \end{matrix})$

role="math" localid="1647934685591" $(\begin{matrix} 2 n \\ n \end{matrix}) = (\begin{matrix} n \\ 0 \end{matrix}) (\begin{matrix} n \\ n \end{matrix}) + (\begin{matrix} n \\ 1 \end{matrix}) (\begin{matrix} n \\ n - 1 \end{matrix}) + . . . . + (\begin{matrix} n \\ n \end{matrix}) (\begin{matrix} n \\ 0 \end{matrix})$ .......................... (2)

We know that,

${}^{n}C_{r} = {}^{n}C_{n - r}$ ............................. (3)

Substituting (3) in (2), we get

$(\begin{matrix} 2 n \\ n \end{matrix}) = {(\begin{matrix} n \\ 0 \end{matrix})}^{2} + {(\begin{matrix} n \\ 1 \end{matrix})}^{2} + . . . . + {(\begin{matrix} n \\ n \end{matrix})}^{2}$

Therefore, it is proved that $(\begin{matrix} 2 n \\ n \end{matrix}) = {鈭�}_{k = 0}^{n} {(\begin{matrix} n \\ k \end{matrix})}^{2}$ .

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91影视

Use Theoretical Exercise 8 to prove that
$(\begin{matrix} 2 n \\ n \end{matrix}) = {鈭�}_{k = 0}^{n} {(\begin{matrix} n \\ k \end{matrix})}^{2}$

Short Answer

Step by step solution

Step 1. State the Theoretical Exercise 8.

Step 2. Prove the given equation.

One App. One Place for Learning.

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91影视

Use Theoretical Exercise 8 to prove that2nn=鈭�k=0nnk2

Short Answer

Step by step solution

Step 1. State the Theoretical Exercise 8.

Step 2. Prove the given equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Logic and Functions

Statistics

Theoretical and Mathematical Physics

Decision Maths

Calculus

Study anywhere. Anytime. Across all devices.

Use Theoretical Exercise 8 to prove that
$(\begin{matrix} 2 n \\ n \end{matrix}) = {鈭�}_{k = 0}^{n} {(\begin{matrix} n \\ k \end{matrix})}^{2}$