Answer
$12,600$ different vertical arrangements.
Work Step by Step
It is a Permutations of 10 flags (four of a kind, three of a second kind, two of a third kind and one of a fourth kind):
$\frac{10!}{4!\times3!\times2!\times1!}=\frac{10\times9\times8\times7\times6\times5\times4\times3\times2\times1}{24\times6\times2\times1}=12,600$
