Question

• A. The quantity on the left is greater
• B. The quantity on the right is greater
• C. Both are equal
• D. The relationship cannot be determined without further information

Hint:

Compulsory items reduce the number of choices you need to make and also reduce the pool of items you can choose from. This always results in fewer combinations compared to a situation without such constraints.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem involves calculating the number of ways a student can choose questions from two independent sections of a paper. The total number of choices is the product of the choices available in each section. The key difference is the constraint of compulsory questions in Column A.
Step 2: Key Formula or Approach:
The number of ways to choose \(r\) questions from a set of \(n\) is the combination \(C(n, r)\). We use the multiplication principle to combine the choices from the two sections.
Step 3: Detailed Explanation:
For Column A:
The student has to make choices in two sections. - Section 1: Attempt 3 questions out of 4. \[ \text{Choices for Section 1} = C(4, 3) = \frac{4!}{3!1!} = 4 \] - Section 2: Attempt 4 questions out of 5, but the first two are compulsory. This means 2 questions are already selected. The student only needs to choose \(4 - 2 = 2\) more questions. The pool of available questions to choose from is also reduced to \(5 - 2 = 3\) questions. \[ \text{Choices for Section 2} = C(3, 2) = \frac{3!}{2!1!} = 3 \] - Total Choices: The total number of ways is the product of choices for each section. \[ \text{Total Choices for A} = 4 \times 3 = 12 \] For Column B:
The student has to make choices in two sections without any compulsory questions. - Section 1: Attempt 3 questions out of 4. \[ \text{Choices for Section 1} = C(4, 3) = 4 \] - Section 2: Attempt 4 questions out of 5. \[ \text{Choices for Section 2} = C(5, 4) = \frac{5!}{4!1!} = 5 \] - Total Choices: The total number of ways is the product of choices for each section. \[ \text{Total Choices for B} = 4 \times 5 = 20 \] Step 4: Final Answer:
Comparing the two quantities:
Quantity A = 12
Quantity B = 20
Quantity B is greater than Quantity A.