Sum of multiplication of all combination of m element from an array of n elements

Question

Suppose I have an array {1, 2, 5, 4} and m = 3. I need to find:

1*2*5 + 1*2*4 + 1*5*4 + 2*5*4

i.e Sum of multipli

Answer 1

This problem is equivalent to calculation of Mth coefficient of polynom with given roots (Vieta's theorem). Adapted algorithm in Delphi (O(N) memory and O(N^2) time):

 function CalcMultiComb(const A: array of Integer; const M: Integer): Integer;
  var
    i, k, N: Integer;
    Coeff: TArray;
  begin
    N := Length(A);
    if (N = 0) or (M > N) then
      Exit;
    SetLength(Coeff, N + 1);

    Coeff[0] := -A[0];
    Coeff[1] := 1;
    for k := 2 to N do begin
      Coeff[k] := 1;
      for i := k - 2 downto 0 do
        Coeff[i + 1] := Coeff[i] - A[k-1] * Coeff[i + 1];
      Coeff[0] := -A[k-1] * Coeff[0];
    end;

    Result := Coeff[N - M];
    if Odd(N - M) then
      Result := - Result;
  end;

calls CalcMultiComb([1, 2, 3, 4], M) with M=1..4 give results 10, 35, 50, 24