Why doesn't the factorial value fit even in QWord?

Question

Pascal

Need to solve a problem outside the formula C=N!/M!*(NM)!. With a value of N=21, the value of N! does not fit anywhere, tried QWord, integer, longint, word, int64, longword errors 201 and 215. N and M are less than 50000.
Code:

function Prost(a:QWord):boolean;
var i:int64;
    f:boolean;
begin
if a<2 then f:=false
else
 begin
  f:=true;
  i:=2;
  while (i*i<=a) and f do
  if a mod i=0 then f:=false
  else i:=i+1;
 end;
 Prost:=f;
end;
function fact(k: QWord) : QWord;
var
  p: QWord;
  i: integer;
begin
    p:=1;
    for i:=2 to k do
    begin
       p:=p*i;
  	End;
  	fact:= p;
end;
var
  N, M, C, count : QWord;
  i : integer;
begin
  read(N);
  read(M);
  count:= 0;
  C:= fact(N) div (fact(M) * fact(N - M));
  for i:= 1 to C do
  Begin
    if C mod i = 0 then
    Begin
      if Prost(i) then inc(count);
    End;
  End;
  write(count);
End.

Answer 1

Long arithmetic is used to store very large numbers , I think you can easily google ready-made libraries.
But something tells me that in your case you need some kind of tricky algorithmic solution, because the factorial of 50000 is a dozen or two seconds on average hardware.

Answer 2

If you look at this formula, you can see that the entire factorial does not need to be calculated and it is slightly reduced (the beginning of the factorial N is divided by the entire factorial M).

import math

def combinations(n, m):
    k = n - m
    if k == 0:
        return 1

    result = 1
    for i in range(m + 1, n + 1):
        result *= i

    result //= math.factorial(k)
    return result

print(combinations(30, 25)) # 142506