<p>Find the number of seven letter words that can be formed using the letters of the word SUCCESS so that the two C's are together but no two S are together.</p>
Step-by-Step Solution
Key Concept: Treat the two C's as a single unit, then arrange this unit with other letters while ensuring the three S's are separated using the gaps method (place non-S letters first, then insert S's in gaps).
<p><strong>Step 1:</strong> Treat the two C's as a single unit since they must be together. We now have: CC (as 1 unit), U, E, S, S, S (three separate S's).</p><p><strong>Step 2:</strong> To ensure no two S's are together, first arrange the non-S letters: the block CC, U, and E. These 3 objects can be arranged in 3! = 6 ways.</p><p><strong>Step 3:</strong> When 3 objects are arranged in a line, they create 4 gaps (before the first, between each pair, and after the last): _O_O_O_</p><p><strong>Step 4:</strong> We need to place 3 identical S's in these 4 gaps such that no two S's are in the same gap (to keep them separated). This is equivalent to choosing 3 gaps from 4 available gaps: C(4,3) = 4 ways.</p><p><strong>Step 5:</strong> Total arrangements = (Arrangements of CC, U, E) × (Ways to place 3 S's) = 6 × 4 = 24</p><p>∴ Answer: <strong>24</strong></p>
Correct Answer: 24