Let $S=\{1,2,3,4,5,6,7,8,9,10\}$. Define$f:S\to S$ as  $f\left(n\right)=\left\{\begin{array}{cc}2n,& \text{ if }n=1,2,3,4,5\\ 2n-11& \text{ if }n=6,7,8,9,10\end{array}\right\}$
Let  g :  S $\to$S be a function such that $fog\left(n\right)=\left(\begin{array}{ll}n+1& ,\text{ if }n\text{ is odd }\\ n-1& ,\text{ if }n\text{ iseven }\end{array}\right)$, 
then $g(10)(g(1)+g(2)+g(3)+g(4)+g(5)$ is equal to 

. $\because f(n)=\{\begin{array}{ll}2n& ,n=1,2,3,4,5\\ 2n-11,& n=6,7,8,9,10\end{array}$

$$\begin{array}{rl}\therefore & f(1)=2,f(2)=4,\dots ,f(5)=10\\ & \text{ and }f(6)=1,f(7)=3,f(8)=5,\dots ,f(10)=9\end{array}$$

Now, $f(g(n))=\{\begin{array}{ll}n+1,& \text{ if }n\text{ is odd }\\ n-1,& \text{ if }n\text{ is even }\end{array}$

$\begin{array}{llr}\therefore & f\left(g\right(10\left)\right)=9\Rightarrow g\left(10\right)=f^{-1}\left(9\right)\Rightarrow g\left(10\right)=10& \\ & f\left(g\right(1\left)\right)=2 \Rightarrow g\left(1\right)=f^{-1}\left(2\right)\Rightarrow g\left(1\right)=1& \\ & f\left(g\right(2\left)\right)=1 \Rightarrow g\left(2\right)=f^{-1}\left(1\right)\Rightarrow g\left(2\right)=6& \\ & f\left(g\right(3\left)\right)=4 \Rightarrow g\left(3\right)=f^{-1}\left(4\right)\Rightarrow g\left(3\right)=2& \\ & f\left(g\right(4\left)\right)=3 \Rightarrow g\left(4\right)=f^{-1}\left(3\right)\Rightarrow g\left(4\right)=7& \\ & f\left(g\right(5\left)\right)=6 \Rightarrow g\left(5\right)=f^{-1}\left(6\right)\Rightarrow g\left(5\right)=3& \\ \therefore & g\left(10\right)\left(g\right(1)+g(2)+g(3)+g(4)+g(5\left)\right)=190& \end{array}$