Question

Let `y` be an element of the set `A={1,2,3,4,5,6,10,15,30}` and `x_(1)`, `x_(2)`, `x_(3)` be integers such that `x_(1)x_(2)x_(3)=y`, then the number of positive integral solutions of `x_(1)x_(2)x_(3)=y` is
A. `81`
B. `64`
C. `72`
D. `90`

Answer 1

Correct Answer - B
`(b)` Number of solutions of the equation `x_(1)x_(2)x_(3)=y` is the same as the number of solutions of equation
`x_(1)x_(2)x_(3)x_(4)=30=2xx3xx5`
where `x_(4)` is dummy variable
Now number of solutions `=` number of ways distinct integers `2`, `3` and `5` can be distributed in four boxes `x_(1)`,`x_(2)`,`x_(3)` and `x_(4)=4^(3)=64`