Question

A student is asked to answer 10 out of 13 questions in an examination such that he must answer at least four questions from the first five questions. Then the total number of possible choices available to him is

• A. 186
• B. 176
• C. 286
• D. 196

Answer 1

The Correct Option is D

Let $N = 10$ be the number of questions to be answered.
Let $T = 13$ be the total number of questions.
The first five questions are denoted by $Q_1, Q_2, Q_3, Q_4, Q_5$.
The remaining 8 questions are denoted by $Q_6, Q_7, \ldots, Q_{13}$.
The student must answer at least 4 questions from the first five questions.
The number of questions chosen from the first 5 questions can be 4 or 5.

Case 1: 
The student answers 4 questions from the first 5 questions.
The student must choose 4 questions from the first 5, which can be done in $\binom{5}{4}$ ways.
Then the student must choose $10 - 4 = 6$ questions from the remaining $13 - 5 = 8$ questions.
This can be done in $\binom{8}{6}$ ways.
So the number of choices in this case is:
$\binom{5}{4} \binom{8}{6} = 5 \cdot \frac{8 \cdot 7}{2 \cdot 1} = 5 \cdot 28 = 140$.

Case 2: 
The student answers 5 questions from the first 5 questions.
The student must choose 5 questions from the first 5, which can be done in $\binom{5}{5}$ ways.
Then the student must choose $10 - 5 = 5$ questions from the remaining $13 - 5 = 8$ questions.
This can be done in $\binom{8}{5}$ ways.
So the number of choices in this case is:

$\binom{5}{5} \binom{8}{5} = 1 \cdot \frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1} = 1 \cdot 56 = 56$.

Total Number of Choices:
The total number of possible choices available to the student is:

$140 + 56 = 196$.

Final Answer:
The final answer is ${196}$.