Question

In an exam:
- Three subjects have maximum marks = $ n $ each
- Fourth subject has max marks = $ 2n $
- Find the number of ways to score total $ 3n $ marks.

• A. \( \frac{1}{6}(n + 1)^2(5n^2 + 10n + 6)^2 \)
• B. \( \frac{1}{6}(n + 1)(5n^2 + 10n + 6)^2 \)
• C. \( \frac{1}{6}(n + 1)^2(5n^2 + 10n + 6) \)
• D. \( \frac{1}{6}(n + 1)(5n^2 + 10n + 6) \)

Hint:

Use combinatorial counting or generating function techniques for bounded integer solution problems.

Answer 1

The Correct Option is D

Let \( x_1 + x_2 + x_3 + x_4 = 3n \), where:
- \( 0 \leq x_1, x_2, x_3 \leq n \)
- \( 0 \leq x_4 \leq 2n \)
We are to count the number of non-negative integer solutions under these bounds. This is a known restricted integer partition problem where:
- 3 variables range over \( [0, n] \)
- 1 variable over \( [0, 2n] \)
- Sum = \( 3n \)
The result for such problems is derived using generating functions or inclusion-exclusion. The known result is: \[ \boxed{ \frac{1}{6}(n + 1)(5n^2 + 10n + 6) } \]