## Relationships Between Coefficients of the Binomial Expansion

The binomial expansion is
$(a+b)^n={}^nC_0a^n+{}^nC_1a^{n-1}b+{}^nC_2a^{n-2}b^2+...+{}^nC_{n-2}b^2a^{n-2}+{}^nC_{n-1}ab^{n-1}+{}^nC_nb^n$
The
${}^nC_k$
are the coefficients of the binomial expansion, and are also sometimes written
$\begin{pmatrix}n\\k\end{pmatrix}$
.
${}^nC_k=\frac{n!}{k!(n-k)!}$
.
The
${}^nC_k$
are not isolated. There are relationships between coefficients.
${}^nC_k={}^nC_{n-k}$
- coefficients are symmetric.
${}^5C_2=\frac{5!}{2!(5-2)!}=\frac{5!}{2!3!}=\frac{5!}{3!2!}={}^5C_3$
.
$\sum_{k=0}^n {}^nC_k=\sum_{k=0}^n {}^nC_{n-k}=2^n$
- the sum of the binomial coefficients is
$2^n$
.
$(a+b)^3=a^3+3a^2b+3ab^2+b^3$
.
The sum of the coefficients is
$1+3+3+1=8=2^3$
.
$\frac{{}^nC_k+1}{{}^nC_k}=\frac{n!/((k+1)(n-(k+1))}{n!/(k!(n-k)!}=\frac{n-k}{k+1} \rightarrow {}^C_{k+1}=\frac{n-k}{k+1}{}^nC_k$
.
${}^8C_6=28=\frac{8-6}{5+1} \times 56=\frac{8-6}{5+1} \times {}^8C_5$

Also,
$0!=1!=1$