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  4. Total number of positive integral solutions of the equation x1 ⋅ x2 ⋅ x3-60, is equal to

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Q. Total number of positive integral solutions of the equation $x_{1} \cdot x_{2} \cdot x_{3}-60$, is equal to

Permutations and Combinations

Solution:

Considering the equation $x_{1} x_{2} x_{3}=60=(2)^{2}$. (3) $^{1}$. (5)
Let $x_{i}=2^{\alpha i}, 3^{\beta i}, 4^{\gamma i}, \alpha i, \beta i, \gamma i \geq 0$
$\Rightarrow (2)^{\alpha_{i}+\alpha_{2}+\alpha_{3}} \cdot(3)^{\beta_{i}+\beta_{2}+\beta_{3}} \cdot(5)^{\gamma_{1}+\gamma_{2}+\gamma_{3}} \cdot(2)^{2} \cdot(3)^{1} \cdot(5)^{1} $
$ \Rightarrow \alpha_{1}+\alpha_{2}+\alpha_{3}=2 ; \beta_{1}+\beta_{2}+\beta_{3}=1 ; \gamma_{1}+\gamma_{2}+\gamma_{3}=1$
Total number of positive integral solutions =
${ }^{4} C _{2} \cdot{ }^{3} C _{2} \cdot{ }^{3} C _{2}=6 \times 3 \times 3=54$