How to count the number of nonnegative integer solutions to $x_1 + x_2 + x_3 + x_4 = 21$ such that $x_1$, $x_2$, $x_3$, $x_4 ≤ 7$
2026-04-01 13:11:08.1775049068
Count the numbers of integer solutions of the ecuation $x_1 + x_2 + x_3 + x_4 = 21$
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If you think about it you can see that you want the coefficient of $x^{21}$ in the expansion of $(1+x+x^2+\cdots + x^7)^4.$ I don't know an easy way to get that, but on a symbolic algebra calculator the coefficient is $120.$