Question:
The total number of positive integral solutions (x, y, z) such that xyz=24 is :
The total number of positive integral solutions (x, y, z) such that xyz=24 is :
Show Hint
To find the number of ways to distribute $n$ identical items into $r$ distinct bins, use the formula $^{n+r-1}C_{r-1}$.
Updated On: Feb 12, 2026
- 24
- 30
- 36
- 45
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The Correct Option is B
Solution and Explanation
Step 1: $24 = 2^3 \times 3^1$.
Step 2: For powers of 2: $x=2^{a_1}, y=2^{a_2}, z=2^{a_3} \Rightarrow a_1+a_2+a_3=3$. Ways = $^{3+3-1}C_{3-1} = ^5C_2 = 10$.
Step 3: For powers of 3: $b_1+b_2+b_3=1$. Ways = $^{1+3-1}C_{3-1} = ^3C_2 = 3$.
Step 4: Total = $10 \times 3 = 30$.
Step 2: For powers of 2: $x=2^{a_1}, y=2^{a_2}, z=2^{a_3} \Rightarrow a_1+a_2+a_3=3$. Ways = $^{3+3-1}C_{3-1} = ^5C_2 = 10$.
Step 3: For powers of 3: $b_1+b_2+b_3=1$. Ways = $^{1+3-1}C_{3-1} = ^3C_2 = 3$.
Step 4: Total = $10 \times 3 = 30$.
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