Question

How does Pascal's triangle relate to binomial expansion?

Answer 1

It tells you the coefficients of the terms.

Explanation:

Let's consider the `n-th` power of the binomial `(a+b)`, namely `(a+b)^n`. It must be a polynomial in `a` and `b` of degree `n`, and so every term must be of degree `n`, which means that the exponents of `a` and `b` must sum to `n`. Let's make a couple of examples:

• `(a+b)^2 = a^2+2ab+b^2`. All terms are of degree two: The exponent of `a^2` is `2`, and the same goes for `b^2`, while in `2ab`, we have `ab=a^1b^1`, and so again `1+1=2`.

• `(a+b)^3 = a^3 + 3a^2b + 3ab^2+b^3`, and all terms are either cubic (`a^3` and `b^3`), or the exponents of the variables sum up to three: `a^2b` and `ab^2` lead to `1+2=2+1=3`.

So, when expanding the power of a binomial, you must count how many possible combinations you have to find numbers `i` and `j` such that `i+j=n`. These numbers will be the exponents of the variables, and you will consider the sum of `a^ib^j` with some coefficients. And here comes Pascal's triangle. It tells you the coefficients of the progressive terms in the expansions.

For example, the first line of the triangle is a simple `color(green)1`. And indeed, `(a+b)^0=color(green)1`.

The second line is `color(green)1 \ \ color(red)1`. And in fact, `(a+b)^1 = color(green)1a+ color(red)1 b`.

The third line is `color(green)1 \ \ color(blue) 2\ \ color(red)1`. And as we've seen above,
`(a+b)^2=color(green)1a^2 +\ \ color(blue) 2ab\ \+ color(red)1b^2`.

And so on: if you look above, you have that the coefficients of the cubic expansion are `1\ \ 3\ \ 3\ \ 1`, which is exactly the fourth line of the triangle.