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Chapter 7: Problem 45

Find the ninth row of Pascal's triangle.

Short Answer

Expert verified
The ninth row of Pascal's triangle is: \(1, 8, 28, 56, 70, 56, 28, 8, 1\).

Step by step solution

01

Understand Pascal's Triangle Structure

Pascal's triangle is a symmetric triangle, starting from the top with one number, which is 1. Each row of the triangle has one more number than the row above it, and the numbers on the outer edges are always 1.
02

Find the First Eight Rows of the Triangle

Before finding the ninth row, it's helpful to build the first eight rows of Pascal's triangle, using the rule that each number is the sum of the two numbers directly above it: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1
03

Identify the Rule for Finding the Numbers in the Ninth Row

To find the numbers in the ninth row, recall that each number is the sum of the two numbers directly above it. Since we already have the first eight rows, we can find the ninth row by adding the appropriate numbers from the eighth row.
04

Calculate Each Number in the Ninth Row

Using the rule identified in Step 3, find each number in the ninth row: 1 8 28 56 70 56 28 8 1 To check that this is correct, let's make sure that the ninth row has ten numbers, is symmetric, and has 1's at both ends.
05

Verify the Results Are Correct

Examine the calculated ninth row of Pascal's triangle to ensure it meets the requirements: 1 8 28 56 70 56 28 8 1 This row has ten numbers, is symmetric, and has 1's at both ends. Therefore, the ninth row of Pascal's triangle is: 1, 8, 28, 56, 70, 56, 28, 8, 1

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