Permutations & Combinations
Arrangements
Grade None
Question:
<p>From six different novels and three different dictionaries, four novels and one dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is</p>
<p>(1) less than 500.</p>
<p>(2) at least 500 but less than 750.</p>
<p>(3) at least 750 but less than 1000.</p>
<p>(4) at least 1000.</p>
Step-by-Step Solution
Key Concept: The dictionary must occupy the fixed middle position (position 3 out of 5), so you only select and arrange novels in the remaining 4 positions, and the dictionary selection is independent.
<p><strong>Step 1:</strong> We need to select 4 novels from 6 and 1 dictionary from 3, then arrange them with the dictionary fixed in the middle position.</p><p><strong>Step 2:</strong> Since the dictionary must be in the middle (position 3 of 5 positions), we only arrange the 4 novels in the remaining 4 positions (1, 2, 4, 5).</p><p><strong>Step 3:</strong> Number of ways to select 4 novels from 6: C(6,4) = 15</p><p><strong>Step 4:</strong> Number of ways to arrange these 4 novels in 4 fixed positions: 4! = 24</p><p><strong>Step 5:</strong> Number of ways to select 1 dictionary from 3: C(3,1) = 3</p><p><strong>Step 6:</strong> Total arrangements = C(6,4) × 4! × C(3,1) = 15 × 24 × 3 = 1080</p><p>∴ Answer: 1080</p>
Correct Answer: 4