Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

a) What is Pascal’s triangle?
b) How can a row of Pascal’s triangle be produced from the one above it?

(a) A geometric arrangement of the binomial coefficients in a triangle which is based on Pascal's identity $n \geq k, (\begin{matrix} n + 1 \\ k \end{matrix}) = (\begin{matrix} n \\ k \end{matrix}) + (\begin{matrix} n \\ k - 1 \end{matrix})$ is known as Pascal’s triangle.

(b) A row of Pascal’s triangle can be produced from the one above it by using the formula $(\begin{matrix} n \\ k - 1 \end{matrix}) + (\begin{matrix} n \\ k \end{matrix}) = (\begin{matrix} n + 1 \\ k \end{matrix})$ .

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Concept Introduction

Pascal's triangle is a triangular array of binomial coefficients found in probability theory, combinatorics, and algebra in mathematics.

Step 2: Pascal’s Triangle

(a)

Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle based on the principle called Pascal's identity which states that for $n \geq k, (\begin{matrix} n + 1 \\ k \end{matrix}) = (\begin{matrix} n \\ k \end{matrix}) + (\begin{matrix} n \\ k - 1 \end{matrix})$ . The row in the triangle consists of the binomial coefficients –

$(\begin{matrix} n \\ k \end{matrix}), k = 0, 1, 2, \dots, n$

Therefore, Pascal triangle is based on Pascal’s identity $n \geq k, (\begin{matrix} n + 1 \\ k \end{matrix}) = (\begin{matrix} n \\ k \end{matrix}) + (\begin{matrix} n \\ k - 1 \end{matrix})$ .

Step 3: Row of Pascal’s Triangle

(b)

The $k^{th}$ term in the ${(n + 1)}^{th}$ row in Pascal's triangle is just the sum of the ${(k - 1)}^{th}$ and $k^{th}$ terms in its above row, i.e., the $n^{th}$ row. Let $T_{k} (n)$ be the $k^{th}$ term in the $n^{th}$ row of Pascal's triangle for $k = 0, 1, 2, . . ., n,$ then the ${(n + 1)}^{th}$ row can be obtained as –

$T_{k} (n + 1) = T_{k - 1} (n) + T_{k} (n) = (\begin{array}{c} n \\ k - 1 \end{array}) + (\begin{matrix} n \\ k \end{matrix}) = (\begin{array}{c} n + 1 \\ k \end{array})$

Therefore, the row can be obtained using the formula role="math" localid="1668683483371" $(\begin{matrix} n \\ k - 1 \end{matrix}) + (\begin{matrix} n \\ k \end{matrix}) = (\begin{matrix} n + 1 \\ k \end{matrix})$ .

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Discrete Mathematics and its Applications

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Short Answer

a) What is Pascal’s triangle?
b) How can a row of Pascal’s triangle be produced from the one above it?

Step by Step Solution

Step 1: Concept Introduction

Step 2: Pascal’s Triangle

Step 3: Row of Pascal’s Triangle

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Discrete Mathematics and its Applications

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a) What is Pascal’s triangle? b) How can a row of Pascal’s triangle be produced from the one above it?

Step by Step Solution

Step 1: Concept Introduction

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Step 3: Row of Pascal’s Triangle

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a) What is Pascal’s triangle?
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