<p>A telegraph has a number of arms and each arm is capable of taking 4 distinct positions, including the position of rest. If 1023 different signals can be sent in all then find the number of arms.</p>
Step-by-Step Solution
Key Concept: Each arm can be in 4 positions, so n arms give 4^n total configurations. Since rest position must be excluded (it sends no signal), we have 4^n - 1 = 1023 distinct signals.
<p><strong>Step 1:</strong> Set up the problem. Each arm has 4 positions (including rest). With n arms, total configurations = 4^n.</p><p><strong>Step 2:</strong> Exclude the invalid configuration. The configuration where all arms are at rest sends no signal, so valid signals = 4^n - 1.</p><p><strong>Step 3:</strong> Solve for n: 4^n - 1 = 1023</p><p>4^n = 1024</p><p>4^n = 4^5</p><p>n = 5</p><p><strong>Verification:</strong> 4^5 = 1024, so 1024 - 1 = 1023 ✓</p><p>∴ <strong>Answer: 5 arms</strong></p>
Correct Answer: 5