TITLE: You have to estimate $\binom{63}{19}$ in $2$ minutes to save your life.
QUESTION [43 upvotes]: This is from the lecture notes in this course of discrete mathematics I am following. 
The professor is writing about how fast binomial coefficients grow.

So, suppose you had 2 minutes to save your life and had to estimate, up to a factor of $100$, the value of, say, $\binom{63}{19}$. How would you do it? I will leave this (hopefully intriguing!) question hanging and maybe come back to the topic of efficiently estimating binomial coefficients later.

Any ideas/hints on how to do it?

REPLY [2 votes]: So we want to estimate:
$$\frac{63!}{19!44!} $$
We do this by doing a geometric average approximation, replacing each factorial by it's geometric average, and the following approximations $9.5^{19} \approx 9^{20}$ and since it's a matter of life and death we expose we know the 10% and 5% interest tables by heart (useful since $22 = 20\cdot 1.10$ and $31.5 = 30\cdot 1.05$ so the exponents will copy over to the "percentage" part):
$$\frac{31.5^{63}}{ 9.5^{19} \cdot 22^{44}} = \frac{22\cdot 3^{63}\cdot10^{63}}{66\cdot 2^{44}\cdot10^{44}\cdot 3^{2\cdot 20}} = \frac{1}{3}\frac{3^{23}10^{19}}{2^{44}}$$ then cancelling by using $2^3 \approx 3^2 \approx 10$: 
$$ \approx \frac{10^{19}}{2^{11}} \approx 2.5\cdot 10^{15}$$