Question

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

• A. 36
• B. 24
• C. 45
• D. 30

Answer 1

The Correct Option is D

To find the total number of positive integral solutions for the equation \(xyz = 24\), we need to explore the factors of 24 with three positive integer variables.

First, perform the prime factorization of 24:

24 can be expressed as 

\[2^3 \times 3^1\]

.

Now, we need to distribute these prime factors among \(x\), \(y\), and \(z\).

Using a stars and bars approach (distribution of indistinguishable objects into distinguishable boxes):

1. Distribute \(2^3\): We distribute 3 objects (the three 2's) into 3 distinct boxes (x, y, and z):

The formula for this is \(\binom{n+k-1}{k-1}\), where n is the number of objects, and k is the number of boxes. For \(n = 3\) and \(k = 3\),

\(\binom{3+3-1}{3-1} = \binom{5}{2} = 10\).

1. Distribute \(3^1\): We distribute 1 object (the one 3) into 3 distinct boxes:

\(\binom{1+3-1}{3-1} = \binom{3}{2} = 3\).

Thus, the total number of ways to assign these factors to x, y, z is the product of the outcomes of these distributions:

\(10 \times 3 = 30\).

Therefore, the total number of positive integral solutions for the equation \(xyz = 24\) is 30.