Question

The total number of positive integral solutions \((x,y,z)\) of \[ xyz=24 \] is

• A. \(24\)
• B. \(30\)
• C. \(36\)
• D. \(32\)

Hint:

If \[ xyz=n \] and \[ n=p_1^{a_1}p_2^{a_2}\cdots, \] then distribute each prime exponent among the variables using stars and bars, and multiply the resulting counts.

Answer 1

The Correct Option is B

Step 1: Prime-factorise and count distributions.
\(24=2^3\cdot 3^1\). Distribute powers of 2 among \(x,y,z\): solutions to \(a+b+c=3\) with \(a,b,c\geq0\) is \(\binom{5}{2}=10\). Distribute power of 3: solutions to \(a+b+c=1\) is \(\binom{3}{2}=3\).

Step 2: Multiply the counts.
Total ordered triples \(=10\times3=30\). \[ \boxed{30} \]