There are 15 bulbs in a room. Each one of them can be operated independently . The number of ways in which the room can be lighted is

Options

A. `8^(5)-1`
B. `(32)^(2)-1`
C. `(32)^(3)-1`
D. `8^(4)-1`

Answer

To solve the problem of how many ways the room can be lighted with 15 independent bulbs, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
Each bulb can either be ON or OFF. Therefore, for each bulb, there are 2 choices.

2. **Calculating Total Combinations**:
Since there are 15 bulbs and each can be independently operated, the total number of combinations of ON and OFF states for all bulbs is given by:
\[ \text{Total combinations} = 2^{15} \]

3. **Excluding the All OFF State**:
The problem specifies that we want to know the number of ways the room can be lighted, which means we need to exclude the scenario where all bulbs are OFF. Thus, we subtract 1 from the total combinations:
\[ \text{Ways to light the room} = 2^{15} - 1 \]

4. **Calculating \(2^{15}\)**:
We can calculate \(2^{15}\):
\[ 2^{15} = 32768 \]

5. **Final Calculation**:
Now, we subtract 1 to find the number of ways to light the room:
\[ \text{Ways to light the room} = 32768 - 1 = 32767 \]

### Final Answer:
The number of ways in which the room can be lighted is **32767**.

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