$$\left\{\begin{eqnarray*}{} x^5+y^5+z^5+u^5+v^5&=&(x+y+z+u+v)^4\\ x^4+y^4+z^4+u^4+v^4&=&(x+y+z+u+v)^3\\ x^3+y^3+z^3+u^3+v^3&=&(x+y+z+u+v)^2\\ xyzuv&=&1 \end{eqnarray*}\right.$$ Find the number of sets of solutions $(x, y, z, u, v)\in\mathbb R^5$ that satisfy the system of equations.
This is the multinomial formula $$(x_1+x_2+⋯+x_m)^n=∑_{k_1+k_2+⋯+k_m=n}{n \choose k_1,k_2,…,k_m}∏^m_{i=1}x_i^{k_i},$$
where
$${n \choose k_1,k_2,…,k_m}=\frac{n!}{k_1!k_2!⋯k_m!}$$
This only helps for the expansion of the RHS of the equations, but does not help with the question. I do not know how to continue from this point.