There are 10 true-false questions in an examination. Then these questions can…

Concept — the fundamental principle of counting: if a task is completed in r independent stages and every stage offers the same n choices, with the same choices available each time (repetition allowed), then the number of complete outcomes is n × n × ... × n taken r times, that is nr. Each outcome is one full selection: exactly one choice recorded at every stage.

Application to this examination:

1. One stage is the answering of one question. The paper has 10 true-false questions, so the number of stages is r = 10.

2. Choices at one stage: a true-false question can be marked in exactly two ways, True or False, so n = 2.

3. The stages are independent: how one question is marked puts no restriction on any other question, and both labels stay available every time, so repetition is allowed.

4. Apply the rule: total number of ways = 2 × 2 × ... × 2 with ten factors = 210.

5. Evaluate the power: 210 = 1024. The 10 questions can therefore be answered in 1024 ways.

Cross-check: split the power — 210 = (25)2 = 322 = 1024, which agrees. Structural check on a smaller case: for just 2 such questions the rule predicts 22 = 4 outcomes, and listing them gives TT, TF, FT, FF — exactly 4 complete answer patterns, confirming that the rule counts full answer patterns rather than individual marks.

Why the other listed values do not arise:

• 10 × 2 = 20 adds up one choice per question separately; it counts the individual marks available across the paper, not the complete answer patterns.

• 102 = 100 interchanges base and exponent — it would count 10 choices at each of 2 stages, not 2 choices at each of 10 stages.

• 240 = 10 × 4! combines the question count with the arrangements of four distinct objects; this item never asks for orderings of distinct objects.