A dimensionless quantity is constructed in terms of electronic charge $$e$$, permittivity of free space $$\varepsilon_0$$, Planck's constant $$h$$, and speed of light $$c$$. If the dimensionless quantity is written as $$e^{\alpha} \varepsilon_0^{\beta} h^{\gamma} c^{\delta}$$ and $$n$$ is a non-zero integer, then $$(\alpha, \beta, \gamma, \delta)$$ is given by

A dimensionless quantity has zero net power of every base dimension $$\left(M,\,L,\,T,\,I\right)$$.
We therefore write $$e^{\alpha}\,\varepsilon_0^{\beta}\,h^{\gamma}\,c^{\delta}$$ and equate the exponents of each base dimension to zero.

Dimensions of the individual quantities
Electronic charge : $$[e]=I\,T$$
Permittivity of free space : $$[\varepsilon_0]=M^{-1}L^{-3}T^{4}I^{2}$$ (from Coulomb’s law)
Planck’s constant   : $$[h]=M\,L^{2}T^{-1}$$
Speed of light    : $$[c]=L\,T^{-1}$$

Total dimensional exponents after raising to respective powers
$$\begin{aligned} M:&\; -\beta+\gamma \\[2pt] L:&\; -3\beta+2\gamma+\delta \\[2pt] T:&\; \alpha+4\beta-\gamma-\delta \\[2pt] I:&\; \alpha+2\beta \end{aligned}$$

For the product to be dimensionless, each of the four sums must vanish:

$$-\beta+\gamma=0 \;\Longrightarrow\; \gamma=\beta$$

$$-3\beta+2\gamma+\delta=0 \;\Longrightarrow\; -3\beta+2\beta+\delta=0 \;\Longrightarrow\; \delta=\beta$$

$$\alpha+2\beta=0 \;\Longrightarrow\; \alpha=-2\beta$$

The time exponent yields the same relation $$\alpha+2\beta=0,$$ confirming consistency.

Let $$\beta=-n,$$ where $$n$$ is any non-zero integer (the question specifies an integer multiplier).
Then

$$\alpha=2n,\qquad \gamma=-n,\qquad \delta=-n$$

Thus $$(\alpha,\beta,\gamma,\delta)=(2n,\,-n,\,-n,\,-n),$$ which corresponds to
Option A: $$(2n,\,-n,\,-n,\,-n)$$