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  • The number of positive integral solution of the equation <img alt="\mathrm{x_{1} x_{2} x_{3} x_{4} x_{5}=1050}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bx_%7B1%7D%20x_%7B2%7

The number of positive integral solution of the equation \mathrm{x_{1} x_{2} x_{3} x_{4} x_{5}=1050} is

Option: 1

1800


Option: 2

1600


Option: 3

 1400


Option: 4

none of these


Answers (1)

best_answer

Using prime factorization of 1050 , we can write the given equation as

\mathrm{x_{1} x_{2} x_{3} x_{4} x_{5}=2 \times 3 \times 5^{2} \times 7}

We can assign 2, 3 or 7 to any of 5 variables. We can assign entire \mathrm{5^{2}}  to just one variable in 5 ways or can assign \mathrm{5^{2}=5 \times 5}  to two variables in \mathrm{{ }^{5} C_{2}}  ways. Thus, \mathrm{5^{2}} can be assigned in

\mathrm{{ }^{5} C_{1}+{ }^{5} C_{2}=5+10=15 \text { ways }}

Thus, the required number of solutions is \mathrm{5 \times 5 \times 5 \times 15=1875}.

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