Question

A question paper consists of two sections having respectively 3 and 5 questions. The following note is given on the paper “It is not necessary to attempt all the questions. One question from each section is compulsory”. In how many ways can a candidate select the questions?
(A) 216
(B) 217
(C) 224
(D)225

Answer 1

The total number of questions in section A and section B is 3 and 5 respectively. Use the formula to select m items out of n items, $^{n}{{C}_{m}}$ and calculate the number of ways to select 1 question, 2 questions, and 3 questions out of 3 questions for section A. Similarly, calculate the number of ways to select 1 question, 2 questions, 3 questions, 4 questions, and 5 questions out of 5 questions for section B. Now, calculate the number of ways to select one or more than one question for section A and section B respectively. Use the principle of multiplication, and calculate the total number of ways to select one or more than one question from section A and section B.

Complete step by step solution:
According to the question, we are given that the question paper consists of two sections having respectively 3 and 5 questions. The following note is given on the paper “It is not necessary to attempt all the questions. One question from each section is compulsory”.
The total number of questions in section A = 3 …………………………………………(1)
The total number of questions in section B = 5 …………………………………………(2)
We know the formula to select m items out of n items, $^{n}{{C}_{m}}$ ……………………………………..(3)
There is the only restriction that one question from each section must be attempted. Since there are no restrictions for the maximum number of questions to attempt so, the number of questions to attempt can be any from 1 to 3 and 1 to 5 for section A and section B respectively.
Now, using the above logic and the formula shown in equation (3), we get
The number of ways to select 1 question out of 3 questions from section A = $^{3}{{C}_{1}}=3$ …………………………………(4)
Similarly, using the formula shown in equation (3), we get
The number of ways to select 2 questions out of 3 questions from section A = $^{3}{{C}_{2}}=\dfrac{3\times 2}{2}=3$ …………………………………(5)
Similarly, using the formula shown in equation (3), we get
The number of ways to select 3 questions out of 3 questions from section A = $^{3}{{C}_{3}}=1$ …………………………………(6)
On adding equation (4), equation (5), and equation (6), we get
The number of ways to select one more than one question from section A = $3+3+1=7$ …………………………………………(7)
For section B, let us do the same calculations.
Now, using the formula shown in equation (3), we get
The number of ways to select 1 question out of 5 questions from section B = $^{5}{{C}_{1}}=5$ …………………………………(8)
Similarly, using the formula shown in equation (3), we get
The number of ways to select 2 questions out of 5 questions from section B = $^{5}{{C}_{2}}=\dfrac{5\times 4}{2}=10$ …………………………………(9)
The number of ways to select 3 questions out of 5 questions from section B = $^{5}{{C}_{3}}=\dfrac{5\times 4\times 3}{1\times 2\times 3}=10$ …………………………………(10)
The number of ways to select 4 questions out of 5 questions from section B = $^{5}{{C}_{4}}=\dfrac{5\times 4\times 3\times 2}{1\times 2\times 3\times 4}=5$ …………………………………(11)
The number of ways to select 5 questions out of 5 questions from section B = $^{5}{{C}_{5}}=1$ …………………………………(12)
On adding equation (8), equation (9), equation (10), equation (11), and equation (12), we get
The number of ways to select one or more than one question from section B = $5+10+10+5+1=31$ …………………………………………(13)
From equation (7) and equation (13), we have calculated the number of ways to select one more than one question from section A and section B respectively.
We are asked to find the number of ways to select questions from section A and section B.
Now, using the principle of multiplication, we get
The total number of ways to select one or more than one question from section A and section B = $31\times 7$ = 217.
Therefore, the total number of ways that a candidate can select the questions is 217.
Hence, the correct option is (B).

Note: The above question is related to the binomial distribution, so we can also use the formula ${{2}^{n}}-1$ for calculating the number of ways to select questions. For instance, here for section A, the number of ways to select questions, ${{2}^{3}}-1=7$. Similarly, for section B, the number of ways to select questions, ${{2}^{5}}-1=31$.