Relationships Between Coefficients of the Binomial Expansion
\[(a+b)^n={}^nC_0a^n+{}^nC_1a^{n-1}b+{}^nC_2a^{n-2}b^2+...+{}^nC_{n-2}b^2a^{n-2}+{}^nC_{n-1}ab^{n-1}+{}^nC_nb^n\]
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,
\[0!=1!=1\]
