<p>The letters of the word ASSASSIN are written down at random in a row. The probability that no two S occur together is</p>
<p>(a) \(\frac{1}{35}\)</p>
<p>(b) \(\frac{1}{14}\)</p>
<p>(c) \(\frac{1}{15}\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: Use the complement approach by first arranging the non-S letters, then inserting the S's into the available gaps to ensure separation. The total arrangements must account for repeated letters using the multinomial coefficient.
<p><strong>Step 1:</strong> Count total letters: ASSASSIN has 8 letters with A repeated 4 times, S repeated 3 times, I once, N once.</p><p><strong>Step 2:</strong> Total arrangements = 8!/(4!×3!×1!×1!) = 40320/(24×6) = 280</p><p><strong>Step 3:</strong> For favorable cases (no two S's together), first arrange the non-S letters: A, A, A, A, I, N (6 letters).</p><p>Arrangements of these = 6!/(4!×1!×1!) = 720/24 = 30</p><p><strong>Step 4:</strong> These 6 letters create 7 gaps (before first, between each pair, after last): _A_A_A_A_I_N_</p><p><strong>Step 5:</strong> We need to place 3 S's in these 7 gaps such that no gap has more than 1 S (to keep S's separated).</p><p>Ways to choose 3 gaps from 7 = C(7,3) = 35</p><p><strong>Step 6:</strong> Probability = (30 × 35)/280 = 1050/280 = 15/4 ÷ 4 = <strong>15/28</strong></p><p>∴ Answer: D</p>
Correct Answer: D