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 𝛼ℎ 𝛽𝐺 𝛾 . Which of the following is the correct option? (A) 𝛼 = 7, 𝛽 = −1, 𝛾 = −2 (B) 𝛼 = −7, 𝛽 = −1, 𝛾 = −2 (C) 𝛼 = 7, 𝛽 = −1, 𝛾 = 2 (D) 𝛼 = −7, 𝛽 = 1, 𝛾 = −2

 To determine the correct exponents 

$\alpha$, $\beta$, and $\gamma$ in the equation $Y=c^{\alpha }{h}^{\beta }{G}^{\gamma }$, we need to perform a dimensional analysis. Here's a step-by-step breakdown:

1. Identify the Dimensions of Each Quantity:

   • Young's Modulus ($Y$):

     $$[Y]=\frac{\text{Force}}{\text{Area}}=\frac{ML{T}^{-2}}{L^2}=M{L}^{-1}{T}^{-2}$$

   • Speed of Light ($c$):

     $$[c]=L{T}^{-1}$$

   • Planck's Constant ($h$):

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

   • Gravitational Constant ($G$):

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

2. Set Up the Dimensional Equation:

   $$[Y]=[c{]}^{\alpha }[h{]}^{\beta }[G{]}^{\gamma }$$

   Substituting the dimensions:

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

3. Equate the Exponents for Each Fundamental Quantity:

   • Mass ($M$):

     $$1=\beta -\gamma$$

   • Length ($L$):

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

   • Time ($T$):

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

4. Solve the System of Equations:

   • From the mass equation:

     $$\beta =1+\gamma$$

   • Substitute $\beta =1+\gamma$ into the time equation:

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

   • Substitute $\alpha =-1+3\gamma$ and $\beta =1+\gamma$ into the length equation:

     $$-1=(-1+3\gamma )+2(1+\gamma )+3\gamma ⟹-1=-1+3\gamma +2+2\gamma +3\gamma ⟹-1=1+8\gamma ⟹8\gamma =-2⟹\gamma =-\frac{1}{4}$$

   • However, this leads to a contradiction, indicating a need to re-evaluate the approach.

5. Re-evaluating the Approach:
   Let's consider the provided options and check which one satisfies the dimensional equation.

   • Option (A): $\alpha =7$, $\beta =-1$, $\gamma =-2$

     $$[Y]=c^7{h}^{-1}{G}^{-2}=(L{T}^{-1}{)}^7(M{L}^2{T}^{-1}{)}^{-1}(M^{-1}{L}^3{T}^{-2}{)}^{-2}$$

     Simplifying:

     $$=L^7{T}^{-7}\cdot M^{-1}{L}^{-2}{T}^1\cdot M^2{L}^{-6}{T}^4=M^1{L}^{-1}{T}^{-2}$$

     This matches the dimensions of $Y$.

   • Option (B): $\alpha =-7$, $\beta =-1$, $\gamma =-2$

     $$[Y]=c^{-7}{h}^{-1}{G}^{-2}=(L{T}^{-1}{)}^{-7}(M{L}^2{T}^{-1}{)}^{-1}(M^{-1}{L}^3{T}^{-2}{)}^{-2}$$

     Simplifying:

     $$=L^{-7}{T}^7\cdot M^{-1}{L}^{-2}{T}^1\cdot M^2{L}^{-6}{T}^4=M^1{L}^{-15}{T}^{12}$$

     This does not match the dimensions of $Y$.

   • Option (C): $\alpha =7$, $\beta =-1$, $\gamma =2$

     $$[Y]=c^7{h}^{-1}{G}^2=(L{T}^{-1}{)}^7(M{L}^2{T}^{-1}{)}^{-1}(M^{-1}{L}^3{T}^{-2}{)}^2$$

     Simplifying:

     $$=L^7{T}^{-7}\cdot M^{-1}{L}^{-2}{T}^1\cdot M^{-2}{L}^6{T}^{-4}=M^{-3}{L}^{11}{T}^{-10}$$

     This does not match the dimensions of $Y$.

   • Option (D): $\alpha =-7$, $\beta =1$, $\gamma =-2$

     $$[Y]=c^{-7}{h}^1{G}^{-2}=(L{T}^{-1}{)}^{-7}(M{L}^2{T}^{-1}{)}^1(M^{-1}{L}^3{T}^{-2}{)}^{-2}$$

     Simplifying:

     $$=L^{-7}{T}^7\cdot M^1{L}^2{T}^{-1}\cdot M^2{L}^{-6}{T}^4=M^3{L}^{-11}{T}^{10}$$

     This does not match the dimensions of $Y$.

6. Conclusion:
   Only Option (A) satisfies the dimensional equation for Young's modulus.

Final Answer:

$$\boxed{{\displaystyle A}}$$