$$\because$$ $$f(n) = \left\{ {\matrix{ {2n,} & {n = 1,2,3,4,5} \cr {2n - 11,} & {n = 6,7,8,9,10} \cr} } \right.$$

$$\therefore$$ f(1) = 2, f(2) = 4, ......, f(5) = 10

and f(6) = 1, f(7) = 3, f(8) = 5, ......, f(10) = 9

Now, $$f(g(n)) = \left\{ {\matrix{ {n + 1,} & {if\,n\,is\,odd} \cr {n - 1,} & {if\,n\,is\,even} \cr } } \right.$$

$$\therefore$$ $$\matrix{ {f(g(10)) = 9} & { \Rightarrow g(10) = 10} \cr {f(g(1)) = 2} & { \Rightarrow g(1) = 1} \cr {f(g(2)) = 1} & { \Rightarrow g(2) = 6} \cr {f(g(3)) = 4} & { \Rightarrow g(3) = 2} \cr {f(g(4)) = 3} & { \Rightarrow g(4) = 7} \cr {f(g(5)) = 6} & { \Rightarrow g(5) = 3} \cr } $$

$$\therefore$$ $$g(10)g(1) + g(2) + g(3) + g(4) + g(5)) = 190$$