The document discusses Pascal's triangle and the binomial theorem. It notes that Zhu Shijiei first wrote about the binomial theorem in 1261 and that Pascal expanded on it in 1653 by adding more rows to the triangle. The document then lists five patterns that can be found in Pascal's triangle and the binomial theorem, such as the coefficient of each term and the exponents of a and b relating to the overall exponent n.

1. The Binomial Theorem
or "one of the ways G-d built the universe" ...
Zhu Shijiei 1261
Pascal 1653

2. Expand and simplify ...
a+b

3. Find a pattern, add two more rows to the triangle ...
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1

4. Evaluate each term ...

5. Pascal's Triangle
How many different patterns can you find in the triangle?
SYMMETRY
Counting Numbers
Sum of COUNTING numbers
triangular numbers
Sum of Triangular numbers

6. Pascal's Triangle
How many different patterns can you find in the triangle?

7. Pascal's Triangle
How many different patterns can you find in the triangle?

9. The Binomial Theorem ...
Algebraically
Combinatorically
Notice the patterns ...
(1) The coefficient of the term is:
(2) The exponent on a is given by: [n - (i - 1)]
(3) The exponent on b is given by: i
(4) This relation holds for each term in the expansion:
[exponent on a] + [exponent on b] = n
(5) The number of terms in any binomial expansion is: n + 1