The total number of positive integral solutio | vedclass

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(B) Given $xyz = 24$. Prime factorization of $24$ is $2^3 	imes 3^1$. Let $x = 2^{x_1} 	imes 3^{y_1}$,$y = 2^{x_2} 	imes 3^{y_2}$,and $z = 2^{x_3}

Vedclass · Accepted Answer

$30$

Vedclass · Answer

(B) Given $xyz = 24$. Prime factorization of $24$ is $2^3 	imes 3^1$. Let $x = 2^{x_1} 	imes 3^{y_1}$,$y = 2^{x_2} 	imes 3^{y_2}$,and $z = 2^{x_3} 	imes 3^{y_3}$,where $x_i, y_i \ge 0$. Then $x_1 + x_2 + x_3 = 3$ and $y_1 + y_2 + y_3 = 1$. The number of non-negative integral solutions for $x_1 + x_2 + x_3 = 3$ is given by the formula $\binom{n+r-1}{r-1} = \binom{3+3-1}{3-1} = \binom{5}{2} = 10$. The number of non-negative integral solutions for $y_1 + y_2 + y_3 = 1$ is $\binom{1+3-1}{3-1} = \binom{3}{2} = 3$. Therefore,the total number of positive integral solutions is $10 	imes 3 = 30$.

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

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