Multiple choice

A person goes in for an examination in which there are four papers with a maximum of $m$ marks from each paper. The number of ways in which one can get $2 m$ marks is

  1. $ ^{2m+3}C_3 $
  2. $ \dfrac{1}{3} (m+1)(2m^2+4m+1) $
  3. $ \dfrac{1}{3} (m+1)(2m^2+4m+3) $
  4. $ ^{2m-3}C_3 $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

This is a stars and bars problem equivalent to finding the number of non-negative integer solutions to x1 + x2 + x3 + x4 = 2m, where each xi <= m. Using the inclusion-exclusion principle, the total solutions are C(2m+4-1, 4-1) = C(2m+3, 3), and subtracting cases where at least one xi > m gives the result.

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