Arrangements of 19 balls in three colours

In how many distinguishable orders can $3$ red, $7$ blue and $9$ green balls be placed in a row, where balls of the same colour are identical?

$11{,}085{,}360$. This is a multinomial count: $\tfrac{19!}{3!\,7!\,9!}$, dividing out the rearrangements of identical balls within each colour.