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How many different scalene triangles are possible with a perimeter of 99 units given that the lengths of all the sides are integers? A 175 B 168 C 192 D 196
When the perimeter is odd, the number of scalene triangles possible is <(p-3)^2/48>
When the perimeter is even, the number of scalene triangles possible is <(p-6)^2/48>
Here
In this case, the perimeter is 99 which is odd. Hence, number of scalene triangles possible is <(99-3)^2/48> = 96*96/48 = 192
The maximum length of any side can be 49. (If one side is 50, then the sum of the other two sides, 49, will be less than this side)
Let the lengths of the three sides be 49-x, 49-y, 49-z, where x, y, z can vary from 0 to 48.
Now, since the perimeter is 99, 49 - x + 49 - y + 49 - z = 99 => x+y+z = 48 and 0 <= x, y, z <= 48.
This can be found out by using the number of integral solutions formula. $$^{n+r-1}C_{r-1}$$ = $$^{48+3-1}C_{3-1}$$ = $$^{50}C_2$$ = 49*25 =1225
This is the total number of triangles.
To get scalene triangles, we have to get the number of integral solutions for x+y+z = 48, where 0 <= x < y < z <= 48
If x = y = 0, there is one solution; similarly for x = y = 1 and so on till x = y = 24 -> 25 solutions in all (This also takes care of one equilateral triangle case).
Again, two variables can be equal in three ways - x,y ; y,z or z,x. So, the number of isosceles triangles is 24*3 = 72
Number of equilateral triangles = 1
So, number of scalene triangles = 1225 - 73 = 1152
But, there are for ordered x, y and z. Since we want the general case of unordered variables, we have to divide this by 3! = 6
So, number of scalene triangles =1152/6 = 192
For example, using the above formula if we get the number of triangles as 15.6; Should the answer be 15 or 16? That is, gif or just the closest integer?
