There are 10 lamps in a hall. Each lamp can be switched on or off…
2011
There are 10 lamps in a hall. Each lamp can be switched on or off independently. In how many ways can the hall be illuminated?
Answer: B. 1023 — ConceptWhen n independent objects each have two possible states, the multiplication principle gives 2n total state assignments. If a required condition…
- A.
102
- B.
1023
- C.
210
- D.
10!
Attempted by 7 students.
Show answer & explanation
Correct answer: B
Concept
When n independent objects each have two possible states, the multiplication principle gives 2n total state assignments. If a required condition excludes one configuration, subtract that configuration.
Application
Each of the 10 lamps has two states: on or off.
Therefore, all switching configurations number 210 = 1024.
The all-off configuration does not illuminate the hall, so the required number is 1024 - 1 = 1023.
Cross-check
Choosing at least one lit lamp is the same as choosing a non-empty subset of the 10 lamps. The binomial sum C(10, 1) + C(10, 2) + ... + C(10, 10) = 210 - C(10, 0) = 1023 confirms the result.