1. Consider the equilibrium:

$\mathrm{PCl_5(g)} \rightleftharpoons \mathrm{PCl_3(g)} + \mathrm{Cl_2(g)}$

Here, the number of moles of gaseous species changes as:

$1 \to 2$

So, on the product side there are more moles of gas.

2. Effect of adding helium depends on the condition:

• At constant volume: Helium is an inert gas. Adding it increases the total pressure, but the partial pressures of $\mathrm{PCl_5}$, $\mathrm{PCl_3}$ and $\mathrm{Cl_2}$ remain unchanged because

  $p_i = \frac{n_iRT}{V}$

  and $n_i, T, V$ for reacting gases do not change. Therefore, the reaction quotient remains unchanged, so equilibrium is not affected.

  Hence, Option B is correct in this situation.

• At constant pressure: Adding helium causes the volume to increase. Then partial pressures of all reacting gases decrease proportionally. Since equilibrium shifts toward the side with more moles of gas, the reaction shifts forward:

  $\mathrm{PCl_5} \to \mathrm{PCl_3} + \mathrm{Cl_2}$

  So more $\mathrm{PCl_3}$ and $\mathrm{Cl_2}$ are formed.

  Hence, Option A is correct in this situation.

3. Check the remaining options:

• Option C: False. Helium is inert and does not deactivate $\mathrm{PCl_5}$.
• Option D: False. If volume increases at constant pressure, equilibrium shifts forward, not backward.

4. Therefore, depending on whether helium is added at constant pressure or constant volume, both statements A and B can be correct.

So the correct options are:

$\boxed{A,\ B}$