The time complexity of an algorithm is given by O(n!) then complexity will: Options constant polynomial exponential Factorial

To determine the nature of the time complexity $O(n!)$, let's analyze the provided options.

1. Break Down the Problem

   • We need to categorize $O(n!)$ into one of the given complexity classes: constant, polynomial, exponential, or factorial.

2. Relevant Concepts

   • Constant time complexity $O(1)$ implies that the algorithm runs in a fixed amount of time regardless of input size.
   • Polynomial time complexity, denoted as $O(n^k)$ for some constant $k$, indicates that the time grows polynomially with input size.
   • Exponential time complexity, represented as $O(2^n)$ or similar forms, indicates growth that doubles with each additional input.
   • Factorial time complexity, $O(n!)$, suggests that the time grows factorially with input size $n$.

3. Analysis and Detail

   • Factorial growth ($n!$) is significantly faster than polynomial and exponential growth as $n$ increases.
   • For example:
     • $O(1)$: 1, 1, 1, 1...
     • $O(n)$: 1, 2, 3, 4...
     • $O(n^2)$: 1, 4, 9, 16...
     • $O(2^n)$: 1, 2, 4, 8, 16...
     • $O(n!)$: 1, 2, 6, 24, 120...

4. Verify and Summarize

   • As observed, $O(n!)$ is much larger than exponential complexity. Thus, it belongs in the factorial category.

Final Answer

The time complexity $O(n!)$ will be classified as Factorial.