Stirling's approx for large factorials - need trick please

[mmwave]

To calculate the multiplicities of 600 heads in 1000 coin tosses you start with 1000 choose 600 or
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?

[Hurkyl]

If only there was a way to transform exponentiation into multiplication, and multiplication into addition...

[mmwave]

I tried the log formula for large factorials first but I got numbers I couldn't take the antilog of. But you made me think of taking the logs where I left off and subtracting the log of 2^N then antilog. If that doesn't work then I'm still lost.

[Hurkyl]

Well, the result does have 291 decimal digits.