Relationships Between Binomial Coefficients

The binomial expansion can be written  
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,

The second means that successive binomial coefficients are related.
The first coefficient is  

The second is  

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  
\[6 \times \frac{1}{6}=1\]

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