Question

In an examination, a student has to answer four questions out of five questions. However, questions 1 and 2 are compulsory. Find the number of ways in which students can make a choice.
[a] 6
[b] 3
[c] 2
[d] 60

Answer 1

Hint: Use the fact that the number of ways in which we can select r objects out of n distinct objects is given by $^{n}{{C}_{r}}$. Use bijections principle, which states that if there exists a bijection between two finite sets A and B, then their cardinality is also the same, i.e. if a function $f:A\to B$ exists such that f(x) is both one-one and onto, then n(A) = n(B). Hence find the number of ways in which the student can make a choice.

Complete step-by-step answer:
Since the student has to select Questions 1 and 2, the only choice left for him is to select two questions out of the remaining three questions.
Claim: The number of ways in which the student can select questions is equal to the number of ways of selecting two objects out of 3 distinct objects.
Consider sets A and B, where A is the set of all possible selections of the questions done by the students and B is the set of all possible selection of 2 distinct objects from the objects labelled 3, 4 and 5
Consider a function f from A to B such that f matches question ids with objects having the same labels.
Clearly, f is a bijection, and hence the number of elements in A is the same as that of B
Hence the number of ways in which the student can select the questions is $^{3}{{C}_{2}}=\dfrac{3!}{\left( 3-2 \right)!2!}=3$
Hence the student can select questions in 3 possible ways.
Hence option [b] is correct.

Note: Alternative solution:
The total number of ways in which 4 questions can be selected out of 5 questions ${{=}^{5}}{{C}_{4}}=4$
The number of ways in which Question 1 is not selected ${{=}^{4}}{{C}_{4}}=1$
The number of ways in which Question 2 is not selected ${{=}^{4}}{{C}_{4}}=1$
The number of ways in which neither Question 1 nor Question 2 is selected is 0
Hence by the principle of inclusion and exclusion the total number of ways in which both question 1 and 2 are selected $=5-1-1+0=3$, which is the same as obtained above
Hence option [b] is correct.