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

Hint: First we will find all the possibilities of selecting a number of questions. Here, selecting 5 questions at least from the first 6 means we have to select 5 and 6 number of questions and remaining from leftover questions and add them to find the number of choices available to students.

Complete step-by-step answer:
The student has to solve 10 questions out of 13.
Possibility of choosing the questions are:
He has to choose at least 5 out of the first 6 questions. The student can choose to do all the first 6 questions or only 5 of them.
Students choose to do all the first 6 questions and 4 out of the next 7 questions i.e..,${}^6{C_6} \times {}^7{C_4} = 35$ ways.
Students choose to do 5 out of the first 6 questions and 5 out of the next 7 questions i.e..,${}^6{C_5} \times {}^7{C_5} = 6 \times 21 = 126$ ways.
Hence, total number of ways is 35+126=161
Hence option ‘C’ is the correct answer.

Note: Whenever such type of question is given first write the statement what we have to choose from given question that satisfies the question condition. Then apply the property of choosing a number of ways from a given condition after completing a number of ways which can be selected from 1st condition and 2nd condition and add them both to get the required number of ways.