Permutations & Combinations
Combinations
Grade 11
Question:
<p>In a room, there are 12 bulbs of the same wattage, each having a separate switch. The number of ways to light the room with different amount of illumination is</p>
<p>(1) \(12^2 - 1\)</p>
<p>(2) \(2^{12}\)</p>
<p>(3) \(2^{12} - 1\)</p>
<p>(4) \(12^2\)</p>
Step-by-Step Solution
Key Concept: Each bulb can be independently ON or OFF, creating 2^12 total states. Since 'no illumination' (all OFF) doesn't count as 'lighting the room', we exclude it: 2^12 - 1 ways.
<p><strong>Step 1:</strong> Each of the 12 bulbs has 2 independent states: ON or OFF.</p><p><strong>Step 2:</strong> Total number of ways to set all switches = 2 × 2 × 2 × ... × 2 (12 times) = 2^12 = 4096.</p><p><strong>Step 3:</strong> This includes the case where ALL bulbs are OFF (no illumination). Since we need at least some illumination to 'light the room', we exclude this one case.</p><p><strong>Step 4:</strong> Number of ways to light the room with different illumination = 2^12 - 1 = 4096 - 1 = 4095.</p><p>∴ Answer: C</p>
Correct Answer: C