Expanding and equation

Question

Av looking for a simple way to expand an equation like $(a+b)^6$

This method should help me in doing the same for others with higher exponents.

Answer 1

Are you familiar with binomial coefficients? $$\binom{n}{m}=\frac{n!}{m!(n-m)!}.$$

They represent the number of ways to pick $m$ things from a set of $n$, where you are not interested in the order in which you picked them. They are also the numbers found in Pascal's triangle among many other interesting connections.

They are generated using the relations $ \binom{n}{0}= \binom{n}{n}=1$ for all $n \geq 0$, and $\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}$ for all $0<k<n$.

You can use the binomial theorem to expand $(a+b)^n$ for any $n\in\mathbf{N}$ as:

$$(a+b)^n=\sum_{k=0}^n \binom{n}{k}a^kb^{n-k}.$$

Try to prove it using the information I gave, it's not that hard and will help you understand and remember the formula. It's very useful.

Answer 2

I think Pascal's triangle is what you're probably looking for.