My take on the above question:
For determining the denominator:
100 is to be divided in three non-zero positive variables and as it is positive, minimum value of all three is 1. Therefore, 97 is to be divided amongst three variables now. Place two separator between 97 identical elements and total therefore, total elements are now 99. Position of these seprator can be arranged in 99C2 ways.
For numerator we will have to use the condition x>20, y>10 and z>20. Therefore, taking their minimums as 21, 11 and 21 (totalling to 53), we are left with 47 to distribute among theee variables as earlier. This will give 49C2. So probability will be 49C2/99C2. @chetan2u @bunuel can verify if this makes sense or not
the-number-of-ordered-triplets-of-positive-integers-which-are-the-solu-355796.htmlA similar question for basic understanding