You are given three wires A, B and C of the same length and cross section. They are each stretched by applying the same force to the ends. The wire A is stretched least and comes back to its original length when the stretching force is removed. The wire B is stretched more than A and also comes back to its original length when the stretching force is removed. The wire C is stretched most and remains stretched even when stretching force is removed. The greatest Young’s modulus of elasticity is possessed by the material of wire

The correct option is A.

Let us understand this step-by-step by analyzing the concept of Young's modulus of elasticity.
Young's modulus ($Y$) is defined as the ratio of tensile stress to tensile strain within the elastic limit of the material:

$Y=\frac{\text{Stress}}{\text{Strain}}$

Let the original length of each wire be $L$ and the cross-sectional area be $A$.
Since all three wires have the same length and cross-section, and the same stretching force ($F$) is applied to each of them, the tensile stress applied to all three wires is identical:

$\text{Stress}=\frac{F}{A}=\text{constant}$

The strain is given by the ratio of change in length ($\Delta L$) to the original length ($L$):

$\text{Strain}=\frac{\Delta L}{L}$

Substituting this back into the formula for Young's modulus:

$Y=\frac{F/A}{\Delta L/L}=\frac{F·L}{A·\Delta L}$

Since $F$, $L$, and $A$ are the same for all three wires, we can see that Young's modulus is inversely proportional to the elongation or strain:

$Y\propto \frac{1}{\Delta L}$

This means that for a given applied force, the material that undergoes the least elongation ($\Delta L$) possesses the greatest Young's modulus.
Comparing the three wires within their elastic regions:
- Wire A is stretched the least ($\Delta L_A$ is the smallest) and recovers its original shape, meaning it behaves elastically.
- Wire B is stretched more than A ($\Delta L_B>\Delta L_A$).
- Wire C is stretched the most and undergoes plastic deformation as it does not return to its original length.

Because wire A has the smallest deformation under the same force within its elastic limit, it has the smallest strain. Consequently, wire A has the greatest Young's modulus of elasticity.