Sorry, I was wrong for one of the Jeopardy answers today!

*n*! is **not** in *O*(2^{n}).

It is easy to see that *n*! < *n*^{n}, since all the numbers in the product to compute *n*! are less than (or equal to) *n*, but this doesn’t prove it is not in *O*(2^{n}). For that, we can use Sterling’s approximation which gives a tight approximation of the value of *n*! as

which is definitely not in

*O*(2

^{n}) since the base of the exponent is not a constant.

For details, see this wikipedia page.

Sorry I was confused on this today! I was thinking of Fibonacci, which is approximated by φ^{n} where φ is the golden ratio (1.618…). This is in *O*(2^{n}) since 1.618… < 2.

I hope it didn’t change the outcome of the game. As my penance, maybe I should add a question related to this on the final to redeem myself.