Somewhere I came across formulas to find the number of integral-sided or scalene triangles possible when you have been given the perimeter of the triangle. Below are the formulas I noted down : -
If perimeter is given, then the number of integral sided triangle possible :-
$1. \frac{p^2}{48}$ where $p=$ perimeter and it is even
$2. \frac{(p+3)^2}{48}$ where $p=$ perimeter and it is odd
If perimeter is given, then the number of scalene triangle possible :-
$1. \frac{(p-6)^2}{48}$ where $p=$ perimeter and it is even
$2. \frac{(p-3)^2}{48}$ where $p=$ perimeter and it is odd
I just wanted to validate if these formulas are correct.How the above formulas are derived? Please help me clarify !!!
Thanks in advance !!!
Edit : I was trying to solve this question : <br
How many distinct scalene triangles with integral sides are possible whose perimeter is less than $15$ units?
How can we solve this type of question in a much efficient and easy way? In the above question, the perimeter value is less and thus we can calculate the number of traingles by hit and trial as well but what if the value becomes large, let's say $30$ cm or so.