# Re: st: large numbers in comb(n,k) function: no success

 From Inna Becher To statalist@hsphsun2.harvard.edu Subject Re: st: large numbers in comb(n,k) function: no success Date Thu, 05 Feb 2009 17:40:40 +0100

Thank you, Bill. I've just spoken with a mathematician, he showed me a similar way you did.
```the formula I need is actually:

probability=1- [comb(n-m,k)/comb(n, k)]
so I have useful numbers after the computation. (n=180000, m=2, k=2000)

Inna

William Gould, StataCorp LP schrieb:
```
Inna Becher <inna.becher@uni-konstanz.de> wrote,
I would like to implement a comb(n,k) function. But my Stata does not allow it because of large n, k-numbers. N=180000 and k=2000. Is there any other way to do it? I wasn't successful by using exp(lnfactorial(n)) in mata as well.
```
```
Maarten buis <maartenbuis@yahoo.co.uk> replied,
```The outcome of comb(180000,2000) is going to be,
```
ridiculously large (> 8e+307) and it hits the limit of what can be stored in double precision [...]
```
Yes, that's right.  In fact, the answer is between 1e+4770 and 1e+4771.

comb(n, k) is defined

n!
comb(n, k)  =  ---------   =  (n!)/( k! (n-k)! )
k! (n-k)!

```
Thus,
```       ln(comb(n,k)    =  ln( (n!)/( k! (n-k)! ) )
=  ln(n!) - ln(k!) - ln(n-k)!

```
Stata has a lnfactorial() function, so we can plug in and get
```           . scalar n = 180000

. scalar k = 2000

. display lnfactorial(n) - lnfactorial(k) - lnfactorial(n-k)
10983.753

```
In log base 10, that 10983.753/ln(10) = 4770.1833. Hence my statement, the answer is between 1e+4770 and 1e+4771.
```
```
1e+4770 is unimagineably big. The number of particles in the observable universe is estimated to be between 1e+72 and 1e+87, so it would not be possible to tally 1e+4770.
```
-- Bill
wgould@stata.com
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```
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