Number of different ways to choose questions when a paper has 3 sections of 5 questions each and a student must attempt 4 questions from each section

Introduction / Context:This question is about choosing questions from a structured paper. There are several sections, each with a fixed number of questions, and the student must attempt a given number from each section. The task is to count the number of different sets of questions the student can select.

Given Data / Assumptions:

• The question paper has 3 sections.
• Each section contains 5 distinct questions.
• The student must select exactly 4 questions from each section.
• The order in which the student answers questions does not matter; only which questions are chosen matters.

Concept / Approach:The choice in each section is independent of the others. In each section, the student chooses 4 out of 5 questions, so the count for one section is a combination. To find the total number of ways for all sections together, we multiply the number of choices for each section.

Step-by-Step Solution:Step 1: Consider a single section with 5 questions.Step 2: The number of ways to choose 4 questions from 5 is 5C4.Step 3: Compute 5C4 = 5.Step 4: Since there are 3 sections, each with 5 questions, and the student chooses 4 from each, the same count 5 applies to every section.Step 5: Choices across sections are independent, so total ways = 5 * 5 * 5.Step 6: Compute 5 * 5 * 5 = 125.

Verification / Alternative check:You can think of the total selection as specifying which question is omitted in each section. In each section the student omits exactly 1 question, which can be chosen in 5 ways. Therefore, for 3 sections, the number of ways to select which question is omitted from each section is 5^3 = 125, matching the earlier calculation.

Why Other Options Are Wrong:

• 1000, 625, and 525 are not equal to 5^3 and do not arise from correctly applying the combination and multiplication rules.
• For instance, 625 equals 5^4 and would correspond to an incorrect assumption about four independent choices of 5 options.

Common Pitfalls:One common mistake is to add the choices for each section instead of multiplying them. Another is to confuse the required number of questions per section and attempt to choose 4 questions total across all sections. Always carefully interpret whether the conditions apply separately to each section or to the whole paper.

Final Answer:The student can select questions in 125 different ways, so the correct answer is 125.