# In an examination, a question paper consists of 12 questions divided into two parts i.e.,

Question:

In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?

Solution:

A question paper consists of 12 questions divided into 2 parts, one with 5 and the other with 7 questions.

A student has to attempt 8 questions out of the 12 questions by selecting at least 3 from each part.

$\therefore$ Required number of ways $={ }^{5} C_{3} \times{ }^{7} C_{5}+{ }^{5} C_{4} \times{ }^{7} C_{4}+{ }^{5} C_{5} \times{ }^{7} C_{3}=210+175+35=420$