# Combinations and permutations

``````func P(n, k) { n! / ((n-k)!) }
func C(n, k) { binomial(n, k) }

class Logarithm(value) {
method to_s {
var e = int(value/10.log)
"%.8fE%+d" % (exp(value - e*10.log), e)
}
}

func lstirling(n) {
n < 10 ? (lstirling(n+1) - log(n+1))
: (0.5*log(2*Num.pi*n) + n*log(n/Num.e + 1/(12*Num.e*n)))
}

func P_approx(n, k) {
Logarithm((lstirling(n) - lstirling(n -k)))
}

func C_approx(n, k) {
Logarithm((lstirling(n) - lstirling(n -k) - lstirling(k)))
}

say "=> Exact results:"
for n (1..12) {
var p = n//3
say "P(#{n}, #{p}) = #{P(n, p)}"
}

for n (10..60 `by` 10) {
var p = n//3
say "C(#{n}, #{p}) = #{C(n, p)}"
}

say '';
say "=> Floating point approximations:"
for n ([5, 50, 500, 1000, 5000, 15000]) {
var p = n//3
say "P(#{n}, #{p}) = #{P_approx(n, p)}"
}

for n (100..1000 `by` 100) {
var p = n//3
say "C(#{n}, #{p}) = #{C_approx(n, p)}"
}
``````

#### 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
``````