For equation $$ x_1+x_2+x_3 = 15 $$ Find number of positive integer solutions on conditions: $$ x_1<6, x_2 > 6 $$ Let: $y_1 = x_1, y_2 = x_2 - 6, y_3 = x_3$ than, to solve the problem, equation $y_1+y_2 +y_3 = 9$ where $y_1 < 6,0<y_2, 0<y_3 $ has to be solved. Is this correct?
To solve this equation by inclusion-exclusion, number of solution without restriction have to be found $C_1 (3+9-1,9)$ and this value should be subtracted by $C_2 (3+9-7-1,2)$ , (as the negation of $y_1 < 6$ is $y_1 \geq 7$).
Is this the correct answer ?
Problem must be solved using inclusion-exclusion...