<p>How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?</p>
<p>\(8 \cdot {}^6C_4 \cdot {}^7C_4\)</p>
<p>\(6 \cdot 7 \cdot {}^8C_4\)</p>
<p>\(6 \cdot 8 \cdot {}^7C_4\)</p>
<p>\(7 \cdot {}^6C_4 \cdot {}^8C_4\)</p>
Step-by-Step Solution
Key Concept: Use complementary counting or direct arrangement: first arrange the non-S letters, then insert S's into the gaps created. For MISSISSIPPI, arrange 4 I's, 4 P's, 1 M (which creates 6 gaps), then choose 4 of these 6 gaps for the 4 S's.
<p><strong>Step 1:</strong> Identify the letters in MISSISSIPPI: M(1), I(4), S(4), P(2). Total = 11 letters.</p><p><strong>Step 2:</strong> First, arrange the non-S letters: M, I, I, I, I, P, P. These are 7 letters with I repeated 4 times and P repeated 2 times.</p><p>Arrangements = 7!/(4!×2!) = 5040/(24×2) = 5040/48 = 105</p><p><strong>Step 3:</strong> When 7 letters are arranged in a row, they create 8 possible gaps (before 1st letter, between consecutive letters, and after last letter): _M_I_I_I_I_P_P_</p><p><strong>Step 4:</strong> To ensure no two S's are adjacent, we must place the 4 S's into 4 different gaps from these 8 available gaps.</p><p>Ways to choose 4 gaps from 8 = C(8,4) = 70</p><p><strong>Step 5:</strong> Total arrangements = 105 × 70 = 7350</p><p>∴ Answer: B</p>
Correct Answer: B