A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions.

Correct option: (a) 196

Explanation:

At least 4 from first 5 questions

Total question = 13 and student has to answer 10 questions

∴ He can choose      (4 from first 5 and 6 from 8) or (5 from first 5 and 5 from 8)

∴ Number of ways  = ⁵C₄ + ⁸C₆ + ⁵C₅ + ⁸C₅ 

= (5 × 28) + 56

= 196.

Two cases are possible: 

(i) Selecting 4 out of first five questions and 6 out of remaining 8 questions 

∴ Number of choices in this case = ⁵C₄ × ⁸C₆ = ⁵C₁ × ⁸C₂ = \(\frac{5\times8\times7}{1\times2}=140\) 

(ii) Selecting 5 out of first five questions and 5 out of remaining 8 questions. 

⇒ Number of choices = ⁵C₅ × ⁸C₅ = 1 × ⁸C₃ = \(\frac{5\times8\times7}{1\times2\times3}=56.\) 

∴ Total number of choices = 140 + 56 = 196.