To calculate the multiplicities of 600 heads in 1000 coin tosses you start with 1000 choose 600 or(adsbygoogle = window.adsbygoogle || []).push({});

1000! / (600! * (1000-600)!) which equals 1000! / (600!) * (400!).

Since you can't calculate this easily, apply Stirling's approx.

N! = N^N * e^(-N) * sqrt(2piN). Applying this to numerator and denominator and leaving out sqrt terms since they are not the problem:

1000^1000 * e^(-1000)

--------------------------------

600^600*e-600 * 400^400 *e-400

Now the exponential terms cancel, but you still can't calculate the remaining terms.

Question: What is the trick to reduce this to something I can calculate? Here's my best attempt.

I tried factoring the bases and canceling terms but that gives

10*10 ^1000 * 10^1000 10^1000

--------------------------------------- = --------------------

10*10 ^600 * 10*10 ^400 * 6^400 * 4^400 6^600 * 4 ^400

You can cancel common factor of 2^1000 the same way giving

5^1000 / (3^600 * 2^400)

I still can't calculate this. Is there another approach or am I missing some other opportunity for canceling?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Stirling's approx for large factorials - need trick please

**Physics Forums | Science Articles, Homework Help, Discussion**