Young's modulus of elasticity $Y$ is expressed in terms of three derived quantities, namely, the gravitational constant $G$, Planck's constant $h$ and the speed of light $c$, as $Y = c^\alpha h^\beta G^\gamma$. Which of the following is the correct option?

Step 1: Write the dimensional formulas for all quantities.

Young's modulus of elasticity $Y$ has the same dimensions as stress (force per unit area).

$$[Y] = M L^{-1} T^{-2}$$

The dimensions of the speed of light $c$, Planck's constant $h$, and the gravitational constant $G$ are:

$$[c] = L T^{-1}$$ $$[h] = M L^{2} T^{-1}$$ $$[G] = M^{-1} L^{3} T^{-2}$$

Step 2: Set up the dimensional equation.

Substitute the dimensions into the given relation $Y = c^{\alpha} h^{\beta} G^{\gamma}$.

$$M L^{-1} T^{-2} = (L T^{-1})^{\alpha} (M L^{2} T^{-1})^{\beta} (M^{-1} L^{3} T^{-2})^{\gamma}$$ $$M L^{-1} T^{-2} = M^{\beta - \gamma} L^{\alpha + 2\beta + 3\gamma} T^{-\alpha - \beta - 2\gamma}$$

Step 3: Compare the exponents on both sides.

Equate the powers of $M$, $L$, and $T$.

For $M$:

$$\beta - \gamma = 1$$

For $L$:

$$\alpha + 2\beta + 3\gamma = -1$$

For $T$:

$$-\alpha - \beta - 2\gamma = -2$$

Step 4: Solve the system of equations.

Express $\beta$ in terms of $\gamma$ using the first equation.

$$\beta = 1 + \gamma$$

Substitute this expression for $\beta$ into the third equation to express $\alpha$ in terms of $\gamma$.

$$-\alpha - (1 + \gamma) - 2\gamma = -2$$ $$\alpha = 1 - 3\gamma$$

Substitute the expressions for $\alpha$ and $\beta$ into the second equation.

$$(1 - 3\gamma) + 2(1 + \gamma) + 3\gamma = -1$$ $$1 + 2\gamma = -3$$ $$\gamma = -2$$

Now, calculate the values of $\beta$ and $\alpha$.

$$\beta = 1 + (-2) = -1$$ $$\alpha = 1 - 3(-2) = 7$$

Answer: The correct values are $\alpha = 7$, $\beta = -1$, and $\gamma = -2$.