Three papers each carry a maximum of 100 marks in an examination. A student scores a total of 150 marks. In how many ways N can this happen if he scores at least 60 marks in each of exactly two papers? Find the value of [N/400], where [.] denotes the greatest integer function. (Marks are n

Question

Three papers each carry a maximum of 100 marks in an examination. A student scores a total of 150 marks. In how many ways N can this happen if he scores at least 60 marks in each of exactly two papers? Find the value of [N/400], where [.] denotes the greatest integer function. (Marks are non-negative integers.)

Accepted Answer

3. Case: exactly 2 papers >= 60 (not all 3, since x1+x2+x3=150 with all >= 60 needs >= 180, impossible). Choose which 2 papers: C(3,2)=3 ways. WLOG x1>=60, x2>=60, x3<60. Let a=x1-60, b=x2-60. Then a+b+x3=30, a>=0, b>=0, 0<=x3<=59. Since a+b=30-x3>=0 means x3<=30 (if x3>30 then a+b<0, impossible). Also a<=40, b<=40 (satisfied since a+b<=30). Count: number of non-neg integer solutions to a+b+x3=30 with 0<=x3<=30. For each x3 in {0,...,30}: number of (a,b) solutions = 31-x3... wait, a+b=30-x3, solutions = 31-x3. Total = sumₓ₃₌₀³⁰ (31-x3) = sumₖ₌₁³¹ k = 31*32/2 = 496. Times 3 choices: N = 3*496 = 1488. [N/400] = [1488/400] = [3.72] = 3.

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