Find the total number of positive integral solutions for (x,y,z) such that `xyz=24`.

Options

A. 30
B. 36
C. 90
D. 120

Answer

D

Explanation

We have,
`24=2xx3xx4=2xx2xx6`
`=1xx6xx4=1xx3xx8`
`=1xx2xx12=1xx1xx24`
the number of positive integral solutions of xyz=24 is
`3!+(3!)/(2!)+3!+3!+(3!)/(2!)=30`
and number of integral solutions having two negative factors is `.^(3)C_(2)xx30=90`.
Hence, number of integral solutions=30+90=120