`2^(n) lt n!` holds true for

Options

A. (a) All n
B. (b) `n gt 1`
C. (c) `n gt 3`
D. (d) `n gt 4`

Answer

To determine for which values of \( n \) the inequality \( 2^n < n! \) holds true, we will evaluate the inequality for various integer values of \( n \).

### Step-by-step Solution:

1. **Evaluate for \( n = 1 \)**:
- LHS: \( 2^1 = 2 \)
- RHS: \( 1! = 1 \)
- Comparison: \( 2 \) is not less than \( 1 \) (i.e., \( 2 \not< 1 \)).
- Conclusion: The inequality does not hold for \( n = 1 \).

2. **Evaluate for \( n = 2 \)**:
- LHS: \( 2^2 = 4 \)
- RHS: \( 2! = 2 \)
- Comparison: \( 4 \) is not less than \( 2 \) (i.e., \( 4 \not< 2 \)).
- Conclusion: The inequality does not hold for \( n = 2 \).

3. **Evaluate for \( n = 3 \)**:
- LHS: \( 2^3 = 8 \)
- RHS: \( 3! = 6 \)
- Comparison: \( 8 \) is not less than \( 6 \) (i.e., \( 8 \not< 6 \)).
- Conclusion: The inequality does not hold for \( n = 3 \).

4. **Evaluate for \( n = 4 \)**:
- LHS: \( 2^4 = 16 \)
- RHS: \( 4! = 24 \)
- Comparison: \( 16 < 24 \) (i.e., \( 16 < 24 \)).
- Conclusion: The inequality holds for \( n = 4 \).

5. **Evaluate for \( n = 5 \)**:
- LHS: \( 2^5 = 32 \)
- RHS: \( 5! = 120 \)
- Comparison: \( 32 < 120 \) (i.e., \( 32 < 120 \)).
- Conclusion: The inequality holds for \( n = 5 \).

6. **Evaluate for \( n = 6 \)**:
- LHS: \( 2^6 = 64 \)
- RHS: \( 6! = 720 \)
- Comparison: \( 64 < 720 \) (i.e., \( 64 < 720 \)).
- Conclusion: The inequality holds for \( n = 6 \).

7. **General Observation**:
- As \( n \) increases, \( n! \) grows much faster than \( 2^n \). Therefore, we can conclude that the inequality \( 2^n < n! \) will hold true for all \( n \geq 4 \).

### Final Conclusion:
The inequality \( 2^n < n! \) holds true for \( n > 3 \).