<p>The number of different ways in which five "alike dashes" and eight "alike dots" can be arranged using only seven of these "dashes" and "dots" is</p>
Step-by-Step Solution
Key Concept: We must select exactly 7 items from 5 dashes and 8 dots, then count distinct arrangements. The constraint is we can use at most 5 dashes (since we only have 5 alike dashes available).
<p><strong>Step 1:</strong> We need to select and arrange 7 items total from 5 dashes and 8 dots.</p><p><strong>Step 2:</strong> Let d = number of dashes used, where d ≤ 5. Then dots used = 7 - d.</p><p><strong>Step 3:</strong> We need: d ≤ 5 AND (7 - d) ≤ 8, which gives d ≥ -1. So d ranges from 0 to 5.</p><p><strong>Step 4:</strong> For each valid value of d, the number of arrangements of d dashes and (7-d) dots is C(7, d).</p><p><strong>Step 5:</strong> Total ways = C(7,0) + C(7,1) + C(7,2) + C(7,3) + C(7,4) + C(7,5)</p><p>= 1 + 7 + 21 + 35 + 35 + 21 = 120</p><p>Alternatively: Sum from d=0 to 5 of C(7,d) = (1/2)[2^7 - C(7,6) - C(7,7)] = (1/2)[128 - 7 - 1] = 60... but we include all terms from 0 to 5, which equals 120.</p><p><strong>Step 6:</strong> By symmetry, C(7,k) = C(7,7-k), so C(7,0)...C(7,5) = 1+7+21+35+35+21 = 120.</p><p>∴ Answer: <strong>120</strong></p>
Correct Answer: D