Question

A student has to answer 10 out of 13 questions in an examination. The number of ways in which he can answer if he must answer at least 3 of the first five questions is :
(a) 276
(b) 267
(c) 80
(d) 1200

Answer 1

Hint: For solving this question we will directly use the formula of the number of ways to select $r$ objects from the $n$ distinct objects. After that, we will make three cases. In the first case student answers 3 questions from the first five questions and 7 questions from the last eight questions, in the second case student answers 4 questions from the first five questions and 6 questions from the last eight questions and in the third case, Student answers all the 5 questions from the first five questions and 5 questions from the last eight questions. Then, we will solve for the number of different ways possible for each case separately and add them to get the correct answer.

Complete step-by-step answer:

Given:
A student has to answer 10 out of 13 questions in an examination and he must answer at least 3 questions of the first five question.
Now, before we proceed we should know how to select $r$ objects from the $n$ distinct objects. The formula for the number of different possible ways is ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$ .
Now, we will divide the given question in three cases. The cases are mentioned below:
First case:
Student answers 3 questions from the first five questions and 7 questions from the last eight questions. Then, the number of ways in which he can answer is ${}^{5}{{C}_{3}}\times {}^{8}{{C}_{7}}=\dfrac{5\times 4}{2}\times 8=80$ ways.
Second case:
Student answers 4 questions from the first five questions and 6 questions from the last eight questions. Then, the number of ways in which he can answer is ${}^{5}{{C}_{4}}\times {}^{8}{{C}_{6}}=5\times \dfrac{8\times 7}{2}=140$ ways.
Third case:
Student answers all the 5 questions from the first five questions and 5 questions from the last eight questions. Then, the number of ways in which he can answer is ${}^{5}{{C}_{5}}\times {}^{8}{{C}_{5}}=1\times \dfrac{8\times 7\times 6}{2\times 3}=56$ ways.
Now, to find the number of ways in which he answers at least 3 of the first five questions can be calculated by adding all the above-calculated values which equal to $80+140+56=276$ ways.
Thus, the required number of ways is 276.
Hence, (a) is the correct option.

Note: Here, the student before solving first try to understand the problem and divide the given condition into some cases then, we should apply the formula of the selection of $r$ objects from the $n$ distinct objects correctly and calculate the correct answer. Moreover, don’t forget to include the second and third case because it is given that he has to answer at least 3 questions from the first five questions, not only 3 questions from the first five.