<p>The number of different messages by signals with three dots and two dashes is</p>
Step-by-Step Solution
Key Concept: Each distinct arrangement of 3 identical dots and 2 identical dashes represents a different message. This is a permutation problem with repetition where we need to find the number of ways to arrange n objects where some are identical: P = n!/(n₁!×n₂!).
<p><strong>Step 1:</strong> Identify the problem structure. We have 3 identical dots (•) and 2 identical dashes (–) to arrange in a sequence of 5 positions.</p><p><strong>Step 2:</strong> Apply the formula for permutations with repetition: when arranging n objects where n₁ objects are identical of type 1, n₂ are identical of type 2, etc., the number of arrangements is:</p><p>P = n!/(n₁!×n₂!×...×nₖ!)</p><p><strong>Step 3:</strong> Here, n = 5 (total positions), n₁ = 3 (dots), n₂ = 2 (dashes)</p><p>P = 5!/(3!×2!) = 120/(6×2) = 120/12 = 10</p><p><strong>Step 4:</strong> Examples of distinct messages: •••––, ••–•–, ••––•, •–••–, •–•–•, •––••, –•••–, –••–•, –•–••, ––•••</p><p>∴ <strong>Answer: C (10)</strong></p>
Correct Answer: C