1. We start with the balanced redox reaction :

   
   

   $$2 \, \mathrm{KMnO}_4 + 3 \, \mathrm{H}_2\mathrm{SO}_4 + 5 \, \mathrm{H}_2\mathrm{S} \rightarrow \mathrm{K}_2\mathrm{SO}_4 + 2 \, \mathrm{MnSO}_4 + 5 \, \mathrm{S} + 8 \, \mathrm{H}_2\mathrm{O}$$




2. We want to find out the number of moles of water ($x$) produced and the number of moles of electrons ($y$) involved in this reaction.




3. From the balanced equation, we can see that 8 moles of water are produced from the reaction. So, we can say :

   
   

   $$x = 8$$




4. Hydrogen sulfide ($\mathrm{H}_2\mathrm{S}$) gets oxidized to sulfur ($\mathrm{S}$) in this reaction. Each molecule of $\mathrm{H}_2\mathrm{S}$ loses 2 electrons during this process (as sulfur has an oxidation state of -2 in $\mathrm{H}_2\mathrm{S}$ and 0 in $\mathrm{S}$).




5. So, for every mole of $\mathrm{H}_2\mathrm{S}$, 2 moles of electrons are involved. And since 5 moles of $\mathrm{H}_2\mathrm{S}$ are reacting, the total number of moles of electrons involved is $5 \times 2 = 10$. So, we can say :

   
   

   $$y = 10$$




6. We are asked to find the sum of $x$ and $y$, so we add these two values together to get:

   
   

   $$x + y = 8 + 10 = 18$$

Therefore, the value of $(x + y)$ is 18.