## Relationships Between Binomial Coefficients

The binomial expansion can be written
$(a+b)^n={}^nC_0a^0b^n+{}^nC_1a^1b^{n-1}+...+{}^nC_{n-1}a^{n-1}b^1+{}^nC_0a^nb^0$
.
There are some useful relationships between the coefficients
${}^nC_k=\begin{pmatrix}n\\k\end{pmatrix}=\frac{n!}{k! (n-k)!}$
.
${}^nC_k=\frac{n!}{k! (n-k)!}=\frac{n!}{(n-k)!k!}={}^nC_{n-k}$
.
${}^nC_k=\frac{n!}{k! (n-k)!}=\frac{n-(k-1)}{k} \frac{n!}{(k-1)!(n-(k-1)!}=\frac{n-(k-1)}{k}{}^nC{k-1}$
.
The first of these means the binomial expansion above is symmetric, so that for example,
$(a+b)^6=b^6+6ab^5+15a^2b^4+20a^3b^3+15a^4b^2+6a^5b+a^6$

The second means that successive binomial coefficients are related.
The first coefficient is
$\frac{6!}{0!6!}=1$

The second is
$\frac{6}{1!}=6$

The third is
$6 \times \frac{5}{2}=15$

The fourth is
$15 \times \frac{4}{3}=20$

The fifth is
$20 \times \frac{3}{4}=15$

The sixth is
$15 \times \frac{2}{5}=6$

The seventh is
$6 \times \frac{1}{6}=1$