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6.Permutation and Combination
normal
સમીકરણ $xyz = 90$ ના ધન પૂર્ણાકોની સંખ્યા મેળવો
A
$60$
B
$108$
C
$54$
D
$120$
Solution
$\begin{aligned} xyx &=90=2,3^{2}, 5^{1} \\ \text { Let } x &=2^{ x _{1}} 3^{ y _{1}} 5^{ Z _{1}} \\ y &=2^{ x _{1}} 3^{ y _{2}} 5^{ z _{2}} \\ z &=2^{ x _{3}} 3^{ y _{3}} 5^{ z _{3}} \end{aligned}$
$x _{1}+ x _{2}+ x _{3}=1 \rightarrow\left(\begin{array}{c}1+3-1 \\ 3-1\end{array}\right)=\left(\begin{array}{l}3 \\ 2\end{array}\right)=3$
$y _{1}+ y _{2}+ y _{3}=2 \rightarrow\left(\begin{array}{c}2+3-1 \\ 3-1\end{array}\right)=\left(\begin{array}{c}4 \\ 2\end{array}\right)=6$
$z _{1}+ z _{2}+ z _{3}=1 \rightarrow\left(\begin{array}{c}1+3-1 \\ 3-1\end{array}\right)=\left(\begin{array}{l}3 \\ 2\end{array}\right)=3$
$3,6,3=54 ways$
Standard 11
Mathematics
