Permutations & Combinations
Permutations
Grade None
Question:
<p>From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is</p>
<p>less than 500.</p>
<p>at least 500 but less than 750.</p>
<p>at least 750 but less than 1000.</p>
<p>at least 1000.</p>
Step-by-Step Solution
Key Concept: Fix the dictionary in the middle position first, then select and arrange novels in remaining positions. This reduces the problem to arranging 4 novels from 6 in the 4 remaining slots around the fixed middle position.
<p><strong>Step 1:</strong> Identify the structure. We have 5 items total (4 novels + 1 dictionary) arranged in a row, with the dictionary fixed in the middle (position 3).</p><p><strong>Step 2:</strong> Select 1 dictionary from 3 different dictionaries: <strong>C(3,1) = 3 ways</strong>. This dictionary goes to the middle position (1 way to place it).</p><p><strong>Step 3:</strong> Select 4 novels from 6 different novels: <strong>C(6,4) = 15 ways</strong>.</p><p><strong>Step 4:</strong> Arrange these 4 selected novels in the 4 remaining positions (positions 1, 2, 4, 5): <strong>4! = 24 ways</strong>.</p><p><strong>Step 5:</strong> Apply multiplication principle: Total arrangements = 3 × 15 × 24 = <strong>1080</strong></p><p>∴ Answer: D</p>
Correct Answer: D