Discrete Maths closed

Question

The non-negative integral solutions to the equation

$x_1+x_2+x_3+x_4 \leq10$

 

I got 209 as answer.Is it correct?

Comments

How did you do this?

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Since we want positive integral solutions so it means

$x_1 \geq1, x_2 \geq 1, x_3 \geq 1, x_4 \geq 1$

so inequality transforms to

$x_1+x_2+x_3+x_4 \leq 6$ with conditions $x_1,x_2,x_3,x_4 \geq 0$

Now I break this into 7 parts and add result

$x_1+x_2+x_3+x_4=0\Rightarrow \binom{3}{0} ways$

$x_1+x_2+x_3+x_4=1 \Rightarrow \binom{4}{1} ways$

$x_1+x_2+x_3+x_4=2 \Rightarrow \binom{5}{2} ways$

$x_1+x_2+x_3+x_4=3 \Rightarrow \binom{6}{3} ways$

$x_1+x_2+x_3+x_4=4 \Rightarrow \binom{7}{4} ways$

$x_1+x_2+x_3+x_4=5 \Rightarrow \binom{8}{5} ways$

$x_1+x_2+x_3+x_4=6 \Rightarrow \binom{9}{6} ways$

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i think x>=0 not 1

because it is non negative i.e start from zero

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@Deepanshu-I already did not count that case after which I made $x_i \geq 0$