Question

A student has to answer $10$ out of $13$ questions in an examination choosing at least $5$ questions from the first $6$ questions. The number of choices available to the student is

• A. $63$
• B. $91$
• C. $161$
• D. $196$

Answer 1

The correct option is C $161$

The student has to solve 10 questions out of 13. He has to choose at least 5 out of the first 5 questions.

The student can choose to do all the first 6 questions or only 5 of them.

1. Student chooses to do 6 out of the first 6 questions and 4 out of the next 7 questions.
2. Student chooses to do 5 out of the first 6 questions and 5 out of the next 7 questions.

No of possible ways:

1. $1{\times }_4^{7}C=35$
2. ${}_5^{6}C{\times }_5^{7}C=6\times 21=126$

Hence, total number of ways is $35+126=161$