Q.46 Show that if n and j are integer... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 12: Q.46 (page 837)

Show that if n and j are integers with $0 鈮� j 鈮� n, t h e n (\begin{matrix} n \\ j \end{matrix}) = (\begin{matrix} n \\ n - j \end{matrix})$
Conclude that the Pascal triangle is symmetric with respect
to a vertical line drawn from the topmost entry.

Short Answer

Expert verified

If j and n are integers with $0 鈮� j 鈮� n t h e n (\begin{matrix} n \\ j \end{matrix}) = \frac{n!}{j! (n - j)!}$

Step by step solution

Step 1. Given information

Here we have to show that $f o r 0 鈮� j 鈮� n, t h e n (\begin{matrix} n \\ j \end{matrix}) = (\begin{matrix} n \\ n - j \end{matrix})$

Step 2. Showing nj=nn-j

We know that

$(\begin{matrix} n \\ j \end{matrix}) = \frac{n!}{j! (n - j)!}, j a n d n i n t e g e r s 0 鈮� j 鈮� n t h e n (\begin{matrix} n \\ n - j \end{matrix}) = \frac{n!}{(n - n + j)! (n - j)!} = \frac{n!}{j! (n - j)!} = (\begin{matrix} n \\ j \end{matrix})$

Step 3. Drawing the Pascal triangle

By using $(\begin{matrix} n \\ j \end{matrix})$ ,we can arrange values in a triangular form

$(\begin{matrix} 0 \\ 0 \end{matrix}) (\begin{matrix} 1 \\ 0 \end{matrix}) (\begin{matrix} 1 \\ 1 \end{matrix}) (\begin{matrix} 2 \\ 0 \end{matrix}) (\begin{matrix} 2 \\ 1 \end{matrix}) (\begin{matrix} 2 \\ 2 \end{matrix}) (\begin{matrix} 3 \\ 0 \end{matrix}) (\begin{matrix} 3 \\ 1 \end{matrix}) (\begin{matrix} 3 \\ 2 \end{matrix}) (\begin{matrix} 3 \\ 3 \end{matrix})$

now we can draw Pascal triangle is symmetric with respect

to a vertical line drawn from the topmost entry.

$n = 0, j = 0 1 n = 1, j = 1 1 1 n = 2, j = 2 1 2 1 n = 3, j = 3 1 3 3 1$

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91影视

Show that if n and j are integers with $0 鈮� j 鈮� n, t h e n (\begin{matrix} n \\ j \end{matrix}) = (\begin{matrix} n \\ n - j \end{matrix})$
Conclude that the Pascal triangle is symmetric with respect
to a vertical line drawn from the topmost entry.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Showing nj=nn-j

Step 3. Drawing the Pascal triangle

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91影视

Show that if n and j are integers with 0鈮�j鈮�n,thennj=nn-jConclude that the Pascal triangle is symmetric with respectto a vertical line drawn from the topmost entry.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Showing nj=nn-j

Step 3. Drawing the Pascal triangle

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

Logic and Functions

Discrete Mathematics

Decision Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Show that if n and j are integers with $0 鈮� j 鈮� n, t h e n (\begin{matrix} n \\ j \end{matrix}) = (\begin{matrix} n \\ n - j \end{matrix})$
Conclude that the Pascal triangle is symmetric with respect
to a vertical line drawn from the topmost entry.