If $x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 6$ , then the difference between the number of non negative integral solutions and the number of positive integral solutions will be

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Solution

$x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 6$
Number of non negative integral solutions
$=^{6 + 5 - 1} C_{5 - 1} =^{10} C_{4} = 210$
Number of positive integral solutions
$=^{6 - 1} C_{5 - 1} =^{5} C_{4} = 5$
Difference $= 210 - 5 = 205$

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x_{1} + x_{2} + x_{3} + x_{4} + x_{5} \leq

x_{1} + x_{2} + x_{3} + x_{4} = 20

x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + 10 = 0

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x_{1} + x_{2} + x_{3} + x_{4} = 100

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