Binomial theorem and expanding polynomials

Binomial theorem

\(\left(a+b\right)^n=\Sigma_{k=0}^n\ \binom{n}{k}a^{n-k}b^k\)
\(\binom{n}{k}=\frac{n!}{k!\left(n-k\right)!}\)
Example: \(\left(a+b\right)^4=\Sigma_{k=0}^4\ \binom{4}{k}a^{4-k}b^k=\binom{4}{0}a^4+\binom{4}{1}a^3b^1+\binom{4}{2}a^2b^2+\binom{4}{3}a^1b^3+\binom{4}{4}b^4\)
\(=a^4+4a^3b+6a^2b^2+4ab^3+b^4\)

Pascal's Triangle

Each node represents how many ways you can get to that node
Example: \(1a^4b^0+4a^3b^1+6a^2b^2+4a^1b^3+1a^0b^4\)