Pascals Triangle

Pascal’s Triangle is a triangular array of binomial coefficients, where each number is the sum of the two directly above it.

Properties

1. Symmetry: The triangle is symmetrical.
2. Binomial Coefficients: Each row corresponds to the coefficients of $(a+b{)}^n$.
3. Edges: The edges are always 1.

Pascal’s Triangle

           1
         1   1
       1   2   1
     1   3   3   1
   1   4   6   4   1
 1   5  10  10   5   1

Example: Expand $(a+b{)}^4$

From the 4th row:

$$14641$$

Expansion:

$$(a+b{)}^4=a^4+4a^{3}b+6a^2{b}^2+4a{b}^3+b^4$$

Formula for Binomial Expansion

For $(a+b{)}^n$:

$$(a+b{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})a^{n-k}{b}^k$$

where $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$.

Fun Facts

• The sum of the entries in the $n^{th}$ row is $2^n$.
• The diagonal elements represent Fibonacci numbers.