Question

A question paper consists of $11$ question divided into two section $1$ and $2$. section $1$ consits of $5$ questions and section $2$ consists of 6 questions. In how many ways can a student select $6$ questions, taking at least $2$ questions from each section?

Answer 1

Total number of question =$11$

Part 1 contains$5$ questions and part $2$ contain $6$ questions.

A student has to select $6$ questions all in which there is at least $2$ questions from each part.

Now different possibilities can occur.

Part1 (5 questions) Part2 (6 questions)

$3$ $3$

$4$ $2$

$2$ $4$

Now total number of ways is

= (⁵C₃ × ⁶C₃ )+ ( ⁵C₄ × ⁶C₂ ) + ( ⁵C₂× ⁶C₄ )$

= {(5×4×6×5)/ (2×3×2)+ {(5×6 ×5)/(2)} + {(5×4×6 ×5)/(2×2)}$

= $200$ + $75$ + $150$

= $425$

So, the total number of ways a student can select the questions is $425$

Note: We have to take at least$3$ questions at once, not in different parts like first take$3$, then$2$, etc.

If we do so, then it is wrong because it is given in the question that we have to take at least 2 questions at once.