Permutations & Combinations
Arrangement with constraints
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>at least 750 but less than 1000</p>
<p>at least 1000</p>
<p>less than 500</p>
<p>at least 500 but less than 750</p>
Step-by-Step Solution
Key Concept: Fix the dictionary in the middle position, then select and arrange novels in remaining positions. Since the dictionary placement is fixed, you only need to choose 4 novels from 6 and arrange them in 4 positions around the fixed middle dictionary.
<p><strong>Step 1:</strong> Identify the constraint. The dictionary must be in the middle position of a 5-item arrangement (4 novels + 1 dictionary).</p><p><strong>Step 2:</strong> Select 1 dictionary from 3 dictionaries: <strong>C(3,1) = 3 ways</strong></p><p><strong>Step 3:</strong> Select 4 novels from 6 novels: <strong>C(6,4) = 15 ways</strong></p><p><strong>Step 4:</strong> Arrange the 4 selected novels in 4 positions (2 on left, 2 on right of the fixed middle dictionary): <strong>P(4,4) = 4! = 24 ways</strong></p><p><strong>Step 5:</strong> Apply the multiplication principle: Total arrangements = C(3,1) × C(6,4) × 4!</p><p>= 3 × 15 × 24</p><p>= 3 × 360</p><p>= <strong>1080</strong></p><p>∴ Answer: B (assuming B = 1080)</p>
Correct Answer: B