<p>How many different signals can be given using any number of flags from 5 flags of different colors?</p>
Step-by-Step Solution
Key Concept: Each flag can either be used or not used in a signal, and the order of flags matters. We must count all non-empty subsets and arrange each subset, then sum across all subset sizes.
<p><strong>Step 1:</strong> Signals can be made using 1, 2, 3, 4, or 5 flags (order matters).</p><p><strong>Step 2:</strong> For k flags chosen from 5, number of arrangements = P(5,k) = 5!/(5-k)!</p><p><strong>Step 3:</strong> Calculate each:</p><ul><li>Using 1 flag: P(5,1) = 5</li><li>Using 2 flags: P(5,2) = 5×4 = 20</li><li>Using 3 flags: P(5,3) = 5×4×3 = 60</li><li>Using 4 flags: P(5,4) = 5×4×3×2 = 120</li><li>Using 5 flags: P(5,5) = 5! = 120</li></ul><p><strong>Step 4:</strong> Total = 5 + 20 + 60 + 120 + 120 = <strong>325</strong></p>
Correct Answer: 325