The total number of positive
Question:

The total number of positive integral solutions $(x, y, z)$ such that $x y z=24$ is

1. (1) 36

2. (2) 45

3. (3) 24

4. (4) 30

Correct Option: , 4

Solution:

$x \cdot y \cdot z=24$

$x \cdot y \cdot z=2^{3} \cdot 3^{1}$

Now using beggars method. 3 things to be distributed among 3 persons\

Each may receive none, one or more $\therefore{ }^{5} \mathrm{C}_{2}$ ways

Similarly for $1^{\prime} \therefore{ }^{3} \mathrm{C}_{2}$ ways

Total ways $={ }^{5} \mathrm{C}_{2} \cdot{ }^{3} \mathrm{C}_{2}=30$ ways