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

Explanation

`24=2^(3)xx3`
Now, consider three boxes x,y,z. '3' can be put in any of the three boxes.
Also,2,2,2 can be distributed in the three boxes in ` .^(3+3-1)C_(3-1) = .^(5)C_(2)` ways. Hence, the total number of positive integral solutions is equal to the number of distributions which is given by `3xx .^(5)C_(2)=30`.
Here, it should be noted that if any box remains empty, say x, then x=1.
To find the integral solutions where negative integers are also allowed, any two of the factors in each factorization must be negative.
Now, the number of ways to associate negative sign in each case is `.^(3)C_(2)=3`.
Hence, total number of integral solutions is `30+3xx30=120`.