If mass is written as $$m = kc^{P}G^{-1/2}h^{1/2}$$, then the value of $$P$$ will be : (Constants have their usual meaning with $$k$$ a dimensionless constant)

Given $$m = kc^P G^{-1/2} h^{1/2}$$. Using dimensional analysis:

$$[m] = M$$, $$[c] = LT^{-1}$$, $$[G] = M^{-1}L^3T^{-2}$$, $$[h] = ML^2T^{-1}$$.

$$M = (LT^{-1})^P (M^{-1}L^3T^{-2})^{-1/2} (ML^2T^{-1})^{1/2}$$

$$= L^P T^{-P} \cdot M^{1/2}L^{-3/2}T^{1} \cdot M^{1/2}L^{1}T^{-1/2}$$

$$= M^{1} L^{P-3/2+1} T^{-P+1-1/2}$$

Matching M: $$1 = 1$$ ✓

Matching L: $$0 = P - 1/2 \Rightarrow P = 1/2$$

Matching T: $$0 = -P + 1/2 \Rightarrow P = 1/2$$ ✓

The answer is Option (1): $$\boxed{\frac{1}{2}}$$.