Based on standard formula:
This problem is based on standard results:
No. of non-negative solution for equation x1+x2+x3+-----+xn = r ,where x1,x2...xn>=0 is n+r-1Cr.
Given eqn: x1 + x2 + x3 = 17, here x1,x2,x3>=1. ----------eq(A)
Converting this eqn in standard form as (x1-1) + (x2-1) + (x3-1) = 14 .
Let a = x1-1; b=x2-1; c= x3-1; hence a,b,c >=0.So eqn reduces to a+b+c = 14 where a,b,c>=0 --------eq(B)
Hence no. of positive integral sol for eq(A) is equivalent to no.of non-negative solutions for eq(B).
Hence reqd no. of solutions = n+r-1Cr = 3+14-1C14 = 16C14 = 16C2 = 120 Ans.
P.S: This problem is subset of more generic problem a1x1+a2x2+a3x3+----------------anxn <= r where ki <= ai <= ki