Pascal's Triangle

Algebra-2

1. Fundamental Concepts

Definition: Pascal's Triangle is a triangular array of binomial coefficients where each number is the sum of the two directly above it.
Binomial Coefficients: The coefficients in the expansion of $$(x + y)^n$$ are found in Pascal's Triangle.
Symmetry: Each row of Pascal's Triangle is symmetric.

2. Key Concepts

Basic Rule: $$(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$$

Row Construction: Each row starts and ends with 1, and each interior element is the sum of the two elements directly above it.

Application: Used to expand binomials and solve combinatorial problems.

3. Examples

Example 1 (Basic)

Problem: Expand $$(x + y)^3$$ using Pascal's Triangle.

Step-by-Step Solution:

Identify the row in Pascal's Triangle corresponding to $$n = 3$$: 1, 3, 3, 1
Apply the coefficients to the expansion: $$x^3 + 3x^2y + 3xy^2 + y^3$$

Validation: Substitute $$x = 1$$ and $$y = 1$$ → Original: $$(1 + 1)^3 = 8$$; Simplified: $$1 + 3 + 3 + 1 = 8$$ ✓

Example 2 (Intermediate)

Problem: Find the coefficient of $$x^2y^4$$ in the expansion of $$(x + y)^6$$.

Step-by-Step Solution:

Identify the row in Pascal's Triangle corresponding to $$n = 6$$: 1, 6, 15, 20, 15, 6, 1
The term $$x^2y^4$$ corresponds to the third term from the left (since $$x^2y^4 = \binom{6}{2} x^2 y^4$$).
The coefficient is $$\binom{6}{2} = 15$$.

Validation: Substitute $$x = 1$$ and $$y = 1$$ → Original: $$(1 + 1)^6 = 64$$; Simplified: $$1 + 6 + 15 + 20 + 15 + 6 + 1 = 64$$ ✓

4. Problem-Solving Techniques

Pattern Recognition: Identify patterns in Pascal's Triangle to predict coefficients.
Combinatorial Interpretation: Use the binomial theorem to solve counting problems.
Systematic Expansion: Always start with identifying the correct row in Pascal's Triangle before expanding.