Number of Triangles with Integer sides for a given perimeter.

If the perimeter p is even then, total triangles is [p^2]/48.If the perimeter p is odd then, total triangles is [(p+3)^2]/48If it asks for number of scalene triangle with a given perimeter P, then subtract 6 and apply the same formula . For even [(p-6)^2]/48 and for odd [(p-3)^2]/48.

Where [x] represents neatest integer function. For example [6.7] is 7 not 6 because its nearest integer.

Find the number of triangles with exactly one side odd and perimeter = 203

sum is odd so possibilities are

odd + odd + odd

or

odd + even + even

because one side HAS TO BE odd, its given in the question, so only these 2 possibilities.

But again question says that EXACTLY one side has to be odd .. so we remove the case when all sides are odd.

So total triangles possible - triangles with odd sides

total triangles = (n+3)^2/48 = 884

triangles with sides odd = (101 - 2x) + (101 -2y) + (101-2z) = 203

=> x + y + z = 50

unordered solutions = 234

so 234 triangles have all sides odd..

So total triangles with EXACTLY one side odd = 884 - 234 = 650