If $$E$$, $$L$$, $$M$$ and $$G$$ denote the quantities as energy, angular momentum, mass and constant of gravitation respectively, then the dimensions of $$P$$ in the formula $$P = EL^2M^{-5}G^{-2}$$ are:

We have to find the dimensions of the physical quantity $$P$$ defined through the relation

$$P = E\,L^{\,2}M^{-5}G^{-2}.$$

Here the symbols represent the following physical quantities:

Energy  ⇒  $$E$$ with dimensional formula $$[E]=[ML^{2}T^{-2}]$$

Angular momentum  ⇒  $$L$$ with dimensional formula $$[L]=[ML^{2}T^{-1}]$$ (because angular momentum is the product of moment of inertia $$\bigl[ML^{2}\bigr]$$ and angular velocity $$[T^{-1}]$$)

Mass  ⇒  $$M$$ with dimensional formula $$[M]=[M^{1}]$$

Gravitational constant  ⇒  $$G$$ with dimensional formula $$[G]=[M^{-1}L^{3}T^{-2}]$$ (this follows from Newton’s law $$F = G\frac{m_{1}m_{2}}{r^{2}}$$)

Now we substitute these dimensional expressions into the formula for $$P$$.

First, write the overall dimensional product:

$$[P]=[E]\,[L]^{2}[M]^{-5}[G]^{-2}.$$

Substituting each individual dimension we get

$$[P]=\bigl[ML^{2}T^{-2}\bigr]\;\bigl[ML^{2}T^{-1}\bigr]^{2}\;\bigl[M\bigr]^{-5}\;\bigl[M^{-1}L^{3}T^{-2}\bigr]^{-2}.$$

Now evaluate each factor step by step.

The second factor is a square, so

$$\bigl[ML^{2}T^{-1}\bigr]^{2} = [M^{2}L^{4}T^{-2}].$$

The third factor is simply

$$\bigl[M\bigr]^{-5} = [M^{-5}].$$

The fourth factor has a power $$-2$$, therefore

$$\bigl[M^{-1}L^{3}T^{-2}\bigr]^{-2} = [M^{2}L^{-6}T^{4}].$$

Collecting all four results we have

$$[P] = [ML^{2}T^{-2}]\,[M^{2}L^{4}T^{-2}]\,[M^{-5}]\,[M^{2}L^{-6}T^{4}].$$

Now combine the exponents of each fundamental dimension $$M$$, $$L$$ and $$T$$ separately.

Mass $$(M)$$:

Exponent  $$1 + 2 - 5 + 2 = 0$$

Length $$(L)$$:

Exponent  $$2 + 4 + 0 - 6 = 0$$

Time $$(T)$$:

Exponent  $$-2 - 2 + 0 + 4 = 0$$

Hence the net dimensional formula is

$$[P]=[M^{0}L^{0}T^{0}].$$

This shows that $$P$$ is dimensionless.

Looking at the given options, this matches Option D.

Hence, the correct answer is Option D.