Question

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. The number of choices available to him is equal to?

• A. $140$
• B. $196$
• C. $280$
• D. $346$

Answer 1

The correct option is B

$196$

Explanation for the correct answer:

Step 1:According to the questions.

Total number of questions = $13$

Number of questions student has to answer = $10$

The event is the student has to select at least $4$out of $5$ questions. This situation has two possibilities.

First possibility, student selects $4$ out of $5$ questions and

Second possibility, student selects $5$ out of $5$ questions.

Step 2: Finding the possibility.

Thus number of possibilities, that the student has to select from at least $4$ out of $5$ questions are

$$\begin{gathered} =({}^{5}C_4\times {}^{8}C_6)+({}^{5}C_5\times {}^{8}C_5)...\mathbf{[}\mathbf{\because }{}^{\mathbf{n}}\mathbf{C}_{\mathbf{r}}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{r}\mathbf{!}\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{!}}\mathbf{]} \\ =(5\times 28)+(1\times 56) \\ =140+56 \\ =196 \end{gathered}$$

Hence, option (B) $\mathbf{196}$ is the correct answer.