Assertion (A) : $\Big(\dfrac{1}{5}\Big)^{-5} \times \Big(\dfrac{1}{2}\Big)^{-5} = (10)^{-5}$.

Reason (R) : $p^{-q} = \dfrac{1}{p^q} \text{ and } \dfrac{1}{p^{-q}} = p^q$, p ≠ 0.

1. Both A and R are correct, and R is the correct explanation for A.

2. Both A and R are correct, and R is not the correct explanation for A.

3. A is true, but R is false.

4. A is false, but R is true.

According to assertion : $\Big(\dfrac{1}{5}\Big)^{-5} \times \Big(\dfrac{1}{2}\Big)^{-5} = (10)^{-5}$

Solving L.H.S.,

$\Rightarrow \Big(\dfrac{1}{5}\Big)^{-5} \times \Big(\dfrac{1}{2}\Big)^{-5}\\[1em] \Rightarrow 5^5 \times 2^5 \\[1em] \Rightarrow (5 \times 2)^5 \\[1em] \Rightarrow 10^5.$

Since, L.H.S. ≠ R.H.S.

So, assertion (A) is false.

We know that,

$p^{-q} = \dfrac{1}{p^q} \text{ and } \dfrac{1}{p^{-q}} = p^q$, for p ≠ 0

So, reason (R) is true.

Hence, option 4 is the correct option.