<p>Messages are conveyed by arranging four white, one blue, and three red flags on a pole. Flags of the same color are alike. If a message is transmitted by the order in which the colours are arranged, the total number of messages that can be transmitted if exactly six flags are used is</p>
Step-by-Step Solution
Key Concept: We must select exactly 6 flags from 4 white, 1 blue, and 3 red flags, then count distinct arrangements of these 6 flags where flags of the same color are identical. This requires finding all valid compositions and applying the multinomial coefficient for each.
<p><strong>Step 1: Identify constraints</strong> We use exactly 6 flags from: 4 white (W), 1 blue (B), 3 red (R). Let w, b, r denote counts where w + b + r = 6, w ≤ 4, b ≤ 1, r ≤ 3.</p><p><strong>Step 2: Enumerate valid compositions</strong></p><ul><li>b = 0: w + r = 6, but r ≤ 3, so w ≥ 3. Cases: (w,r) = (3,3), (4,2)</li><li>b = 1: w + r = 5, with w ≤ 4, r ≤ 3. Cases: (w,r) = (2,3), (3,2), (4,1)</li></ul><p><strong>Step 3: Count arrangements for each case</strong> For composition (w,b,r), number of arrangements = 6!/(w!·b!·r!)</p><ul><li>(3,0,3): 6!/(3!·3!) = 720/36 = 20</li><li>(4,0,2): 6!/(4!·2!) = 720/48 = 15</li><li>(2,1,3): 6!/(2!·1!·3!) = 720/12 = 60</li><li>(3,1,2): 6!/(3!·1!·2!) = 720/12 = 60</li><li>(4,1,1): 6!/(4!·1!·1!) = 720/24 = 30</li></ul><p><strong>Step 4: Add all cases</strong> Total = 20 + 15 + 60 + 60 + 30 = <strong>185</strong></p><p>∴ Answer: D (185)</p>
Correct Answer: D