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Yang Hui's (Pascal's) Triangle

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Yang Hui's (Pascal's) Triangle

Study Guide

Key Definition

Yang Hui's Triangle is a triangular arrangement of numbers where each number is the sum of the two directly above it. The formula for finding a number in the triangle is $\binom{n}{r} = \frac{n!}{r!(n-r)!}$.

Important Notes

Each row starts and ends with 1.
The triangle is symmetric.
The $n$th row corresponds to $(x+y)^n$.
The sum of the numbers in the $n$th row is $2^n$.
Numbers in the triangle are coefficients in binomial expansion.

Mathematical Notation

$\binom{n}{r}$ is the binomial coefficient

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems.

Why It Works

Yang Hui's Triangle works for binomial expansions because each number represents a combination, calculated by $\binom{n}{r}$, which is used to determine the coefficients in the expansion of $(x+y)^n$.

Remember

The triangle helps in calculating combinations and expanding binomials with the formula $\binom{n}{r}$.

Quick Reference

Combination Formula:$\binom{n}{r} = \frac{n!}{r!(n-r)!}$

Understanding Yang Hui's (Pascal's) Triangle

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Video explanation of this concept

Beginner

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Beginner Explanation

Each row of Pascal's Triangle starts and ends with 1. Every other entry is the sum of the two numbers directly above it. For example, row 0: 1; row 1: 1 1; row 2: 1 2 1; row 3: 1 3 3 1.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the value in row 4 (n=4) and position 2 (k=2) of Yang Hui's Triangle, assuming both rows and positions start counting from 0?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you have 6 different colored beads and you want to choose 3. How many ways can you choose the beads?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Calculate the sum of the numbers in the 5th row (n=5) of Yang Hui's Triangle, assuming rows start at n=0.

Challenge Quiz

Single Choice Quiz

Advanced

What is the coefficient of $x^3y^2$ in the expansion of $(x+y)^5$?

Recap

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