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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  
\[6 \times \frac{1}{6}=1\]