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how many solutions are there to equation x1+x2+x3 =11

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No solution.

Consider the condition which you have mentioned, let consider the maximum value for each variable

x1 will be at max 1, x2 will be at max 2, and x3 will be at max 3, which all can give maximum (1+2+3) i.e. 6. It will never be 11.

 

Hence number of solution will be 0.
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