title: Time complexity of n factorial
body: <p>I have the following problem but I have no idea how to prove it</p>
  
  <pre><code>n! is
      Select one or more:
   a. O(n^{3}) 
   b. O((n/2)^{n}) 
   c. O((n/3)^{n}) 
   d. O((n+1)!) 
   e. O(n^{n})
   f. O(2^{n})
  </code></pre>
  
  <p>I was thinking of using the property that f(x) ∈ O (g(x)) if and only if there is a value x0 and a constant c>0 such that for all x >= x0 f(x)&lt;=c*g(x)</p>
  
  <p>That would mean for the last one: </p>
  
  <pre><code>n! &lt;= c*2^{n}, for all n&gt;=n0
  n!/2^n &lt;=c, for all n&gt;=n0
  </code></pre>
  
  <p>Is it correct if I state here that the inequality is false as n!/2^n tends to infinity and c is just a constant? </p>
  
  <p>Applying this logic I got that n! is b.,c.,d.,e. but I'm not sure this is correct. </p>