Question Analysis:
In this question, a person is appearing for an examination with four papers, and each paper has a maximum of 20 marks. The task is to find the number of ways in which the person can score a total of 40 marks.

Solution:
To solve this problem, we can use the concept of generating functions. Let's consider each paper as a variable with a maximum value of 20. We need to find the coefficient of the term that represents a total score of 40.

Generating Function:
The generating function for each paper can be represented as:
(1 + x + x^2 + x^3 + ... + x^20)

Total Generating Function:
Since there are four papers, the total generating function would be the product of the generating functions of each paper:
(1 + x + x^2 + x^3 + ... + x^20)^4

Expanding the Generating Function:
To find the coefficient of the term representing a total score of 40, we expand the generating function and look for the term with x^40.

Using Binomial Theorem:
We can use the binomial theorem to expand the generating function. The binomial theorem states that:
(a + b)^n = nC0 * a^n * b^0 + nC1 * a^(n-1) * b^1 + nC2 * a^(n-2) * b^2 + ... + nCn * a^0 * b^n

Calculating the Coefficient:
In our case, a = 1, b = x, and n = 4. We want to find the coefficient of the term with x^40. So, we substitute these values into the binomial theorem:
(1 + x + x^2 + x^3 + ... + x^20)^4 = 1^4 + 4C1 * 1^3 * x + 4C2 * 1^2 * x^2 + 4C3 * 1^1 * x^3 + 4C4 * 1^0 * x^4 + ...

Calculating the Coefficient of x^40:
We only need to consider the terms up to x^40, as the maximum marks for each paper is 20. Therefore, we only need to consider the terms up to the power of 4 in the binomial expansion.

Calculating the Coefficient (continued):
The terms that contribute to the coefficient of x^40 are:
4C0 * 1^4 * x^0 + 4C1 * 1^3 * x^20 + 4C2 * 1^2 * x^20 + 4C3 * 1^1 * x^0 + 4C4 * 1^0 * x^0

Simplifying these terms:
1 + 4 * x^20 + 6 * x^20 + 4 * x^0 + 1 * x^0
= 2 + 10 * x^20

Conclusion:
The coefficient of x^40, which represents a total score of 40, is 0. Therefore, there is no way to score exactly