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Pascal's (Zhu Shijie's) Triangle

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Pascal's (Zhu Shijie's) Triangle

Study Guide

Key Definition

Pascal's Triangle is a triangular arrangement of numbers where each number is the sum of the two directly above it, expressed with $C_{n}^{r} = \frac{n!}{r!(n-r)!}$.

Important Notes

Each row of the triangle corresponds to the coefficients of the binomial expansion of $(x + y)^n$
Numbers on the border of Pascal's Triangle are always $1$
The sum of the numbers in row $n$ is $2^n$
The triangle is symmetrical
Pascal's Triangle can be used to find combinations

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$C_{n}^{r}$ denotes 'n choose r', a binomial coefficient

Remember to use proper notation when solving problems

Why It Works

Pascal's Triangle works because it is based on the principle of combinatorics, where each number in the triangle represents the number of ways to choose elements from a set, expressed using $binomial coefficients$.

Remember

To find a number in Pascal's Triangle, use the formula $C_{n}^{r} = \frac{n!}{r!(n-r)!}$.

Quick Reference

Row Sum:$2^n$

Symmetry:The triangle is symmetrical with respect to its center.

Understanding Pascal's (Zhu Shijie's) Triangle

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Video explanation of this concept

Beginner

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Beginner Explanation

Pascal's Triangle starts with a single 1 at the top. Each new row begins and ends with 1, and every interior number is the sum of the two numbers directly above it from the previous row. This simple rule creates the familiar triangular array.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the sum of the elements in the 4th row of Pascal's Triangle?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine having a bag with different colored marbles. How many ways can you choose 3 marbles out of 7?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If you sum the entries along each shallow diagonal of Pascal's Triangle up to the 10th row, which famous sequence do you obtain?

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following represents the 5th term in the expansion of $(x + y)^6$?

Recap

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