Binomial theorem

The binomial theorem is a theorem in algebra describing the binomial coefficients. It states that

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

$=\binom{n}{0}a^nb^0+\binom{n}{1}a^{n-1}b^1+\cdots+\binom{n}{n-1}a^1b^{n-1}+\binom{n}{n}a^0b^n$

$=a^n+na^{n-1}b+\binom{n}{2}a^{n-2}b^2+\cdots+\binom{n}{n-2}a^2b^{n-2}+nab^{n-1}+b^n$

where $\binom{n}{k}$ is a combination (equal to $\frac{n!}{k!(n-k)!}$).

The binomial theorem is very useful in binomial expansion, as it allows for binomials with large exponents to be expanded with relative ease. For example,

$(a+b)^5=a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5$