How do you prove that $\large{\sum\limits_{(n_1,n_2,n_3),n_1+n_2+n_3=15,n_i\geq0,n_i\in\mathbb{Z}}}P(n_1,n_2,n_3)\times(-1)^{n_3}=(1+1-1)^{15}$?
2026-04-07 14:38:20.1775572700
Sum of polynomial coefficients
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Hint: Consider the expansion of $(x+y-z)^{15}$ as per the multinomial theorem and look at how the multinomial coefficients are expressed. The right choice of $(x, y, z)$ should give you the desired result.