Question:
The number of all possible positive integral solutions of the equation \(xyz = 30\) is:
The number of all possible positive integral solutions of the equation \(xyz = 30\) is:
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Number of positive integral solutions of \(xyz = n\) for \(n = p_1^{a_1} p_2^{a_2} \cdots\) is the product of combinations with repetition.
Updated On: Mar 18, 2026
- 24
- 25
- 26
- 27
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The Correct Option is D
Solution and Explanation
Step 1: Factorize 30
\[ 30 = 2 \times 3 \times 5 \] Step 2: Number of positive integer solutions for \(xyz = 30\)
Number of ordered positive integer solutions equals the number of ways to distribute prime factors among \(x,y,z\). Step 3: Use stars and bars method
Number of solutions: \[ (1+3-1)(1+3-1)(1+3-1) = 3 \times 3 \times 3 = 27 \]
\[ 30 = 2 \times 3 \times 5 \] Step 2: Number of positive integer solutions for \(xyz = 30\)
Number of ordered positive integer solutions equals the number of ways to distribute prime factors among \(x,y,z\). Step 3: Use stars and bars method
Number of solutions: \[ (1+3-1)(1+3-1)(1+3-1) = 3 \times 3 \times 3 = 27 \]
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