A student is to answer 10 out of 13 questions in an examination such t

1.	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. Find the number of choices available to him.
Answer» 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.

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. Find the number of choices available to him.

Answer»

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.