Multiple choice

A test has $4$ parts, the first $3$ parts carry $10$ marks each and the $4^{th}$ part carries $15$ marks. Assuming that marks are not given in the form $p/q\space\left ( p< q, p,q\in I \right ), q\neq 0$ then the number of ways in which the student can get $30$ marks out of $45$ is

  1. $5300$
  2. $711$
  3. $6011$

  4. none of these
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B Correct answer
Explanation

This is a partition problem. We need to find non-negative integer solutions to x1 + x2 + x3 + x4 = 30, with constraints 0 <= xi <= 10 for i=1,2,3 and 0 <= x4 <= 15. The coefficient of x^30 in (1+x+...+x^10)^3 * (1+x+...+x^15) is 711.

AI explanation

The problem requires finding the number of non-negative integer solutions to the equation a + b + c + d = 30, where the variables represent marks from parts carrying 10, 10, 10, and 15 marks respectively. A transformation of variables like p = a/10 and q = d/15 turns this into finding combinations that sum to the required total based on the distinct point values. By calculating the combinations of scores across the four parts that add up to exactly 30, the total number of ways to achieve this score is found to be 711.

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