The question statement is incomplete: it asks for the total number of hydrogen atoms in product A and product B, but the reaction scheme/structures leading to products $A$ and $B$ are not provided.

Without the structures or reaction sequence, the number of hydrogen atoms in $A$ and $B$ cannot be derived from chemistry principles alone.

However, since a stored correct answer is provided as $10$, the only possible comparison I can make is with that stored value.

1. Required information missing:

   • Structure of reactant(s)
   • Reagent(s)/condition(s)
   • Identity of products $A$ and $B$

2. Consequence:

   • It is impossible to independently calculate the total number of hydrogen atoms in $A$ and $B$.

3. Comparison with stored answer:

   • Because the actual scheme is absent, I cannot verify the value rigorously.
   • If forced to report based on the provided stored answer, the answer would be $10$.

So, the problem as shown is underspecified, but using the stored answer gives total hydrogen atoms $=10$.