$m$ be the number of distinct non congruent integer sided triangles each with perimeter $15$

Question

Let $m$ be the number of distinct non congruent integer sided triangles each with perimeter $15$ and $n$ be the number of distinct non congruent integer sided triangles each with perimeter $16$

Then $m-n=?$

a) -2

b) 0

c) 2

d) -4

did not get the question and answer. Thank you for helping.

Answer 1

Brute force method:

Program loops;

function nr(P : integer) : integer;
var
  t,k1,k2,k3 : integer;
  OK : boolean;
begin
  t := 0;
  for k1 := 1 to P do
  begin
    for k2 := k1 to P-k1 do
    begin
      k3 := P - k1 - k2;
      if k3 < k2 then Break;
      OK := (k1 + k2 > k3)
        and (k2 + k3 > k1)
        and (k3 + k1 > k2);
      if not OK then Continue;
      t := t + 1;
      Writeln(k1,' ',k2,' ',k3);
    end;
  end;
  nr := t;
end;

procedure test;
var
  t : integer;
begin
  t := nr(15);
  Writeln; Writeln('m = ',t); Writeln;
  t := nr(16);
  Writeln; Writeln('n = ',t); Writeln;
end;

begin
  test;
end.

Giving answer (c):

1 7 7
2 6 7
3 5 7
3 6 6
4 4 7
4 5 6
5 5 5

m = 7

2 7 7
3 6 7
4 5 7
4 6 6
5 5 6

n = 5

Answer 2

The longest side of a (non-degenerate) triangle must be at least a third the perimeter but less than half the perimeter. For perimeter $15$, this means the longest side, restricted to integer values, is $5$, $6$, or $7$; for perimeter $16$, it's $6$ and $7$. The possibilities, written as $abc$ with $a\ge b\ge c\gt0$, can be listed as

$$771,762,753,744,663,654,\text{ and }555$$ for perimeter $15$, and $$772,763,754,664,\text{ and }655$$ for perimeter $16$. Thus $m=7$ and $n=5$, hence $m-n=2$.

(Note, if you allow degenerate triangles, then you also have $880,871,862,853$, and $844$ for perimeter $16$, but that gives $m-n=-3$, which is not one of the multiple-choice answers.)