In an examination, the maximum marks for each of the four papers namely P, Q, R and S are 100. Marks scored by the students are in integers. A student can score 99% in n different ways. What is the value of n?

The maximum marks for each of the four papers (P, Q, R, S) are 100. Total maximum marks for the examination = 4 papers * 100 marks/paper = 400 marks.

To score 99% in the examination, the student needs to obtain: 99% of 400 marks = (99/100) * 400 = 396 marks.

Let the marks scored in the four papers be p, q, r, and s, respectively. We know that 0 <= p, q, r, s <= 100 (marks are integers and cannot exceed the maximum). The sum of the marks must be 396: p + q + r + s = 396.

Instead of directly solving this equation with upper bounds, it's simpler to consider the "marks lost" from the maximum possible score. The maximum possible score is 100 + 100 + 100 + 100 = 400. The student scored 396 marks. So, the total marks lost by the student across all four papers = 400 - 396 = 4 marks.

Let x1 be the marks lost in paper P (i.e., 100 - p). Let x2 be the marks lost in paper Q (i.e., 100 - q). Let x3 be the marks lost in paper R (i.e., 100 - r). Let x4 be the marks lost in paper S (i.e., 100 - s).

Since 0 <= p, q, r, s <= 100, it implies that 0 <= x1, x2, x3, x4 <= 100. The sum of the marks lost is: x1 + x2 + x3 + x4 = (100 - p) + (100 - q) + (100 - r) + (100 - s) x1 + x2 + x3 + x4 = 400 - (p + q + r + s) x1 + x2 + x3 + x4 = 400 - 396 x1 + x2 + x3 + x4 = 4

We need to find the number of non-negative integer solutions to this equation. The upper bound constraint (x_i <= 100) is automatically satisfied because the sum is only 4, so no individual x_i can exceed 4.

This is a classic stars and bars problem. The number of non-negative integer solutions to x1 + x2 + ... + xk = n is given by the formula (n + k - 1) C (k - 1). Here, n = 4 (the sum of lost marks) and k = 4 (the number of papers/variables).

Number of ways = (4 + 4 - 1) C (4 - 1) = 7 C 3 = (7 * 6 * 5) / (3 * 2 * 1) = 35

Therefore, a student can score 99% in 35 different ways.

The final answer is D.