Explanation
\begin{aligned}
&
\begin{array}
{l}{\mathrm{Given:}^{n}\mathrm{C}_{r-1}:^{n}\mathrm{C}_{r}:^{n}\mathrm{C}_{r+1}} \\
{=1:5:20}
\end{array} \\
& \Rightarrow\frac{n!}{(r-1)!(n-r+1)!}\times\frac{r!(n-r)!}{n!}=\frac{1}{5} \\
& \Rightarrow\frac{r}{(n-r+1)}=\frac{1}{5} \\
& \Rightarrow5r=n-r+1 \\
& \Rightarrow n=6r-1 & & ...(\mathbf{i}) \\
& \mathrm{Also,}\frac{n}{r!(n-r)!}\times\frac{(r+1)!(n-r-1)!}{n!}=\frac{5}{20}=\frac{1}{20} \\
& \Rightarrow\frac{(r+1)}{(n-r)}=\frac{1}{4} \\
& \Rightarrow4r+4=n-r \\
& \Rightarrow n=5r+4 & & ...(\mathrm{ii}) \\
& \text{From (i) and (ii), we get} \\
& 6r-1=5r+4 \\
& \Rightarrow r=5 \\
& \mathrm{So},n=5(5)+4=29 \\
& \mathrm{So,~coefficient~of~}4^{\mathrm{th}}\mathrm{terms}={}^{n}\mathrm{C}_{3}={}^{29}\mathrm{C}_{3} \\
& =\frac{29!}{3!26!}=\frac{29\times28\times27}{3\times2}=3654
\end{aligned}
