right triangle

From Pythagorean theorem（畢氏定理）, we know that the lengths of 1 right triangle should be 3,4 and 5 if the total is 12.

The possiblity to get the lengths as 3,4 and 5 =

=(1/4 * 1) + (1/4 * 2/3) + (1/4 * 1/2)
= 1/4 + 1/6 + 1/8
= 13/24



Explanation for calculation:
There are 4 possible lengths for first side, from 2 to 5;
it can't be 6 or larger as total lengths of any 2 sides should be greater than the third one, if one side is 6, the total of other two sides can only be 6.
It can’t be 1, with the same reason, yon can’t find any combination of the 3 lengths to form a triangle with one side is 1.

The possiblity for first lenght is 3 = 1/4, and then there are 2 possibliy cases from 4 to 5 for the second length
(it can't be 9 or larger, otherwise, total length of 2 side is 12;
it can’t be 3, as the third side will be 6, but as a triangle, the total lengths of any 2 sides should be greater than the third one, otherwise, in this case, length of 2 sides are 3, and the other is 6, that’s a straight line. With the same reason, the second length can’t be 1,2,3,6,7,8,)
So, the possibility of second length is 4 or 5 = 1. So, the possiblity to get right triangle for first side is 3 = 1/4 * 1

The possiblity for first lenght is 4 = 1/4 as well, and then there are 3 possible cases from 3 to 5 for second length, and the possibility of second length is 3 or 5 = 2/3. So, the possiblity to get right triangle for first side is 4 = 1/4 * 2/3.

The possiblity for first lenght is 5 = 1/4 as well, and then there are 4 possibliy cases from 2 to 5, and the possibility of second length is 3 or 4 = 2/4. So, the possiblity to get right triangle for first side is 5 = 1/4 * 1/2.