Chapter 12: Q.46 (page 837)

Show that if n and j are integers with $0 \leq j \leq n, t h e n (\begin{matrix} n \\ j \end{matrix}) = (\begin{matrix} n \\ n - j \end{matrix})$
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 $0 \leq j \leq n t h e n (\begin{matrix} n \\ j \end{matrix}) = \frac{n!}{j! (n - j)!}$

Step by step solution

Step 1. Given information

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

Step 2. Showing nj=nn-j

We know that

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

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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Show that if n and j are integers with $0 \leq j \leq n, t h e n (\begin{matrix} n \\ j \end{matrix}) = (\begin{matrix} n \\ n - j \end{matrix})$
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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Show that if n and j are integers with 0≤j≤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

Calculus

Theoretical and Mathematical Physics

Pure Maths

Mechanics Maths

Discrete Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

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