Number of positive unequal integral solutions of the equation `x+y+z=12` is

Options

A. 21
B. 42
C. 63
D. 84

Answer

B

Explanation

We have, x+y+z=12
Assume `x lt y lt z`. Here, `x,y,z ge1`
`therefore`Solutions of Eq. (i) are
`(1,2,9),(1,3,8),(1,4,7),(1,5,6),(2,3,7),(2,4,6) and (3,4,5)`.
Number of positive integral solutions of Eq. (i)=7 but x,y,z can be arranged in 3!=6
Hence, required number of solutions`=7xx6=42`.