B

Step 1: Understanding the electric flux
Electric flux, \( \Phi_E \), through a surface is given by the formula:
\[ \Phi_E = \vec{E} \cdot \vec{A} \]
where \( \vec{E} \) is the electric field and \( \vec{A} \) is the area vector.

Step 2: Identifying the area vector
For an area of 10 m² lying in the XY plane, the area vector \( \vec{A} \) is directed along the Z-axis (k-direction). Therefore, we can express it as:
\[ \vec{A} = 10 \, \hat{k} \] (since area is in the positive k-direction)

Step 3: Calculating the dot product
The given electric field is:
\[ \vec{E} = 2 \, \hat{i} - 10 \, \hat{j} + 5 \, \hat{k} \]

Now, we calculate the dot product \( \vec{E} \cdot \vec{A} \):
\[ \vec{E} \cdot \vec{A} = \left( 2 \, \hat{i} - 10 \, \hat{j} + 5 \, \hat{k} \right) \cdot \left( 10 \, \hat{k} \right) \]

Since both vectors have only the k-component when considering \( \hat{k} \):
\[ \vec{E} \cdot \vec{A} = 0 + 0 + (5)(10) = 50 \]

Step 4: Conclusion
The electric flux through the surface is:
\[ \Phi_E = 50 \, \text{Nm}^2/\text{C} \]
Therefore, the correct answer to the question is option B.