The number of positive integral solution  of the equation  x1x2x3x4x5=1050 is

# The number of positive integral solution  of the equation  ${x}_{1}{x}_{2}{x}_{3}{x}_{4}{x}_{5}=1050$ is

1. A

1800

2. B

1600

3. C

1400

4. D

none of these

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### Solution:

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

${x}_{1}{x}_{2}{x}_{3}{x}_{4}{x}_{5}=2×3×{5}^{2}×7$

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

ways

Thus, the required number of solutions is 5 × 5 × 5 × 15 = 1875.

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