Question

If P(n) : 2ⁿ < n!
Then the smallest positive integer for which P(n) is true if

• A. 2
• B. 3
• C. 4
• D. 5

Answer 1

The Correct Option is C

If P(n) : \(2^n < n!\), then the smallest positive integer for which P(n) is true if

We test values of \(n\) starting from 1:

• n = 1: \(2^1 < 1!\) => \(2 < 1\) (False)
• n = 2: \(2^2 < 2!\) => \(4 < 2\) (False)
• n = 3: \(2^3 < 3!\) => \(8 < 6\) (False)
• n = 4: \(2^4 < 4!\) => \(16 < 24\) (True)

The smallest positive integer for which P(n) is true is 4.

Answer: (C) 4