The speed of a wave produced in water is given by $$v = \lambda^a g^b \rho^c$$. Where $$\lambda$$, $$g$$ and $$\rho$$ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of $$a$$, $$b$$ and $$c$$ respectively, are

Using dimensional analysis: $$v = \lambda^a g^b \rho^c$$

Dimensions: $$[v] = [LT^{-1}]$$, $$[\lambda] = [L]$$, $$[g] = [LT^{-2}]$$, $$[\rho] = [ML^{-3}]$$

$$[LT^{-1}] = [L]^a [LT^{-2}]^b [ML^{-3}]^c = M^c L^{a+b-3c} T^{-2b}$$

Equating powers:

M: $$c = 0$$

T: $$-1 = -2b$$, so $$b = \frac{1}{2}$$

L: $$1 = a + b - 3c = a + \frac{1}{2}$$, so $$a = \frac{1}{2}$$

$$a = \frac{1}{2}, b = \frac{1}{2}, c = 0$$

This matches option 4.