# Combinations and permutations

Raku can't compute arbitrary large floating point values, thus we will use logarithms, as is often needed when dealing with combinations. We'll also use a Stirling method to approximate ln ⁡ ( n ! ) {\displaystyle \ln(n!)} ![{\displaystyle \ln(n!)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d264f09c80b779e9028658543d5e31ae647cacf1):

ln ⁡ n ! ≈ 1 2 ln ⁡ ( 2 π n ) + n ln ⁡ ( n e + 1 12 e n ) {\displaystyle \ln n!\approx {\frac {1}{2}}\ln(2\pi n)+n\ln \left({\frac {n}{e}}+{\frac {1}{12en}}\right)} ![{\displaystyle \ln n! \approx
\frac{1}{2}\ln(2\pi n) + n\ln\left(\frac{n}{e} + \frac{1}{12 e n}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c30535e750163de96d5626c9d81642fd66739d34)

Notice that Raku can process arbitrary long integers, though. So it's not clear whether using floating points is useful in this case.

```perl
multi P($n, $k) { [*] $n - $k + 1 .. $n }
multi C($n, $k) { P($n, $k) / [*] 1 .. $k }
 
sub lstirling(\n) {
    n < 10 ?? lstirling(n+1) - log(n+1) !!
    .5*log(2*pi*n)+ n*log(n/e+1/(12*e*n))
}
 
role Logarithm {
    method gist {
	my $e = (self/10.log).Int;
	sprintf "%.8fE%+d", exp(self - $e*10.log), $e;
    }
}
multi P($n, $k, :$float!) {
    (lstirling($n) - lstirling($n -$k))
    but Logarithm
}
multi C($n, $k, :$float!) {
    (lstirling($n) - lstirling($n -$k) - lstirling($k))
    but Logarithm
}
 
say "Exact results:";
for 1..12 -> $n {
    my $p = $n div 3;
    say "P($n, $p) = ", P($n, $p);
}
 
for 10, 20 ... 60 -> $n {
    my $p = $n div 3;
    say "C($n, $p) = ", C($n, $p);
}
 
say '';
say "Floating point approximations:";
for 5, 50, 500, 1000, 5000, 15000 -> $n {
    my $p = $n div 3;
    say "P($n, $p) = ", P($n, $p, :float);
}
 
for 100, 200 ... 1000 -> $n {
    my $p = $n div 3;
    say "C($n, $p) = ", C($n, $p, :float);
}
```

#### Output:

```
Exact results:
P(1, 0) = 1
P(2, 0) = 1
P(3, 1) = 3
P(4, 1) = 4
P(5, 1) = 5
P(6, 2) = 30
P(7, 2) = 42
P(8, 2) = 56
P(9, 3) = 504
P(10, 3) = 720
P(11, 3) = 990
P(12, 4) = 11880
C(10, 3) = 120
C(20, 6) = 38760
C(30, 10) = 30045015
C(40, 13) = 12033222880
C(50, 16) = 4923689695575
C(60, 20) = 4191844505805495

Floating point approximations:
P(5, 1) = 5.00000000E+0
P(50, 16) = 1.03017326E+26
P(500, 166) = 3.53487492E+434
P(1000, 333) = 5.96932629E+971
P(5000, 1666) = 6.85674576E+6025
P(15000, 5000) = 9.64985399E+20469
C(100, 33) = 2.94692433E+26
C(200, 66) = 7.26975256E+53
C(300, 100) = 4.15825147E+81
C(400, 133) = 1.25794868E+109
C(500, 166) = 3.92602839E+136
C(600, 200) = 2.50601778E+164
C(700, 233) = 8.10320356E+191
C(800, 266) = 2.64562336E+219
C(900, 300) = 1.74335637E+247
C(1000, 333) = 5.77613455E+274
```
