The number of solutions (x, y, z) to the equation x - y - z = 25, where x, y, and z are positive integers such that x≤40,y≤12, and z≤12 is
x−y−z=25 where x, y, z are positive integers such that x≤40,y≤12,z≤12, y+z=x−25……1
Now, for LHS to have feasible solutions under the given conditions, RHS = (x - 25) can take values from a minimum 2 to the maximum of 15.
Now, let's consider y+z≤15
y+z+k=15…2, where k is any variable.
Putting y=(y′+1),z=(z′+1) in equation 2), we get
Number of solutions for the above equation =(13+3−1)C(3−1)=15C2=15×142=105
But the above number of solutions contains solutions with y′≥12,z′≥12. Those have to removed.
Number of invalid solutions = 2 + 4 = 6 ( 2 solutions for RHS = 13 and 4 solutions for RHS = 12)
Hence, final number of possible solutions = 105 - 6 = 99.