In an examination of Mathematics paper, there are 20 questions of equal marks and the ques…

Question

In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections: A, B and C. A student is required to attempt a total of 15 questions taking at least 4 questions from each section. If section A has 8 questions, section B has 6 questions and section C has 6 questions, then the total number of ways a student can select 15 questions is \_\_\_\_\_

Answer: 11376

Accepted Answer

Let $a,b,c$ be the number of questions selected from sections A, B, C respectively, with $a+b+c=15$, $4\le a\le8$, $4\le b\le6$, $4\le c\le6$.

Summing $\displaystyle\sum \binom{8}{a}\binom{6}{b}\binom{6}{c}$ over all valid $(a,b,c)$ gives the total. Working through the valid combinations (with $a$ ranging $4$ to $8$ subject to $b,c\le6$ and $a+b+c=15$):

$$\text{Total} = 11376$$

Solutions · 1