Explanation:

To solve this analogy, we first need to determine the relationship between the first pair of numbers, $102$ and $10405$. \nStep 1: Analyze the first pair.\nWe observe that $102^2 = 10404$. If we add $1$ to this result, we get $10404 + 1 = 10405$.\nSo, the pattern is $\text{Number}^2 + 1$. \nStep 2: Apply the same pattern to the third number.\nThe third number is $201$.\nFollowing the pattern, we need to calculate $201^2 + 1$. \nStep 3: Calculate $201^2$. $201^2 = (200 + 1)^2 = 200^2 + 2 \times 200 \times 1 + 1^2 = 40000 + 400 + 1 = 40401$. \nStep 4: Add 1 to the result. $40401 + 1 = 40402$. \nStep 5: Evaluate the options.\nOption A: $40402$. This matches our calculated result.\nOption B: $50402$. This is incorrect as it does not follow the established pattern.\nOption C: $40401$. This is $201^2$, but the pattern requires adding $1$ to the square.\nOption D: $50401$. This is incorrect. \nTherefore, the number that replaces the question mark is $40402$.