A teacher set a question paper consist of 8 question of total 40 marks such that each question contains minimum two marks and integer number of marks. In how may ways this can be done?

A. ⁴⁸C₇

B. ³¹C₇

C. ³²C₇

D. ⁴⁰C₈

Solution(By Examveda Team)

40 marks need to divide between 8 questions. Let X₁, X₂, X₃, X₄, X₅, X₆,X₇, X₈.
Now,
X₁ + 2 + X₂ +2 + X₃ +2 + X₄ +2+ X₅ +2 + X₆ + 2+ X₇ +2+ X₈ + 2 = 40.
X₁ + X₂ + X₃ + X₄ + X₅+ X₆+ X₇+ X₈ = 40 - 16
X₁ + X₂ + X₃ + X₄ + X₅+ X₆+ X₇+ X₈ = 24.
Now, we use formula to divide n different things in r part,
= ^{n +r -1}C_{r -1}.
= ^{24 +8 -1}C_{8 -1}.
= ³¹C₇.