The total number of positive integral solutions $( x , y , z )$ such that $xyz =24$ is :
- 36
- 24
- 45
- 30
The Correct Option is D
Solution and Explanation
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):
- 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 \(k = 3\),
\(\binom{3+3-1}{3-1} = \binom{5}{2} = 10\).
- 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.
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