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(D) To find the number of positive integral solutions of $xyz = 60$,we first find the prime factorization of $60$.$60 = 2^2 	imes 3^1 	imes 5^1$.We

Vedclass · Accepted Answer

$54$

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(D) To find the number of positive integral solutions of $xyz = 60$,we first find the prime factorization of $60$.$60 = 2^2 	imes 3^1 	imes 5^1$.We need to distribute the factors $2^2, 3^1, 5^1$ among $x, y, z$.For the factor $2^2$,the number of ways to distribute it among $3$ variables is given by the stars and bars formula $\binom{n+k-1}{k-1}$,where $n=2$ and $k=3$.Number of ways for $2^2 = \binom{2+3-1}{3-1} = \binom{4}{2} = 6$.For the factor $3^1$,the number of ways is $\binom{1+3-1}{3-1} = \binom{3}{2} = 3$.For the factor $5^1$,the number of ways is $\binom{1+3-1}{3-1} = \binom{3}{2} = 3$.Total number of solutions = $6 	imes 3 	imes 3 = 54$.

The number of positive integral solutions of $xyz = 60$ is

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