Pascal's Triangle and Binomial Theorem

Binomial Coefficients

Definition

The coefficient of the -th term of a binomial with exponent (i.e ) is:

Properties

- E.g

Example

In NYC, to get from point A to point B, you must go 9 blocks east, 5 blocks south. Assuming you take a 13 block path, how many paths are there to follow?

Approach 1:

Approach 2:

There are 13 blocks, "chose" 8 of them to be east

Symmetry Theorem

Theorem

(See Choose, Factorial, universally quantified)

Intuition

Since Pascal's Triangle is symmetric, we can go from the left or the right (see below).

Proof

Pascals Triangle

Properties

Let be each column, be each row

Each "cell" is defined as
Each row is symmetrical
- I.e
The sum of each row is

Pascal's Identity

Identity

Different sources will say different things are "Pascal's Identity". Here are some I have come across:

Also

Also

These can all be demonstrated on the triangle

Binomial Theorem

Binomial Theorem 1 (BT1)

Theorem

See union, complex numbers

Binomial Theorem 2 (BT2)

Corollary

Newton's Binomial Theorem

Newton generalized the binomial theorem for real exponents other than non-negative integers (the same also applies to complex exponents).

For this generalization, the finite sum is replaced with an infinite series.

For example:

He noticed that we could generate these expansions with this rule:

where is defined by the recurrence relation:

notice that for , all terms of .

Question

What if is not a positive integer? In that case, the series never terminates.

Example

We with to find the binomial expansion of

solution

Note that we have because our equation is actually .

As a matter of fact, this is exactly what we would get with a Taylor polynomial, centred at :

(see derivative)

However, Newton did now have this tool at his disposal.

How would he check that his result was correct? Newton multiplied both sides by themselves (i.e squared both sides).

Newton did this up to and found that past the first 2 terms, the rest cancelled out. He then assumed, without proof, that the rest of the terms would also cancel. The proof came later.

As such, this generalization became known as Newton's Binomial Theorem.

Note

This is the same as the standard binomial theorem, but generalized to allow .

Examples

Newton specifically wrote the following instances: