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I know if $x_1 + x_2 + x_3 + x_4 \leq k$ then number of integral solution can be given by ${k + 4} \choose 4$ using Stars and Bars but I am unable to figure out no. of integral solutions of $x_1 + x_2^2 + x_3^3 + x_4^4 \leq k$. where $ 0\leq x_1, x_2 ,x_3, x_4 \leq 10^3 $ and $0 \leq k \leq 10^{18}$. Kindly help.

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  • $\begingroup$ The number of nonnegative integer solutions of the inequality $x_1 + x_2 + x_3 + x_4 \leq k$ is the number of nonnegative integer solutions of the equation $x_1 + x_2 + x_3 + x_4 + s = k$, where $s = k - (x_1 + x_2 + x_3 + x_4)$, so you should have obtained $\binom{k + 4}{4}$. $\endgroup$ – N. F. Taussig Jan 13 at 10:02
  • $\begingroup$ sorry. My Bad. Yes it should be ${k + 4} \choose {4} $. I have edited it $\endgroup$ – Brij Raj Kishore Jan 13 at 12:20

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