彩球排成一行

多重集合排列数为

$$\frac{19!}{3!\,7!\,9!}=11085360.$$

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Original Explanation

This is a classic anagrams problem, where we have three distinct groups (corresponding to the three colors) and every string corresponds to an arrangement of red, green, and blue. There are $19$ total balls, so if all were distinct, then there are $19!$ ways to arrange them. However, this overcounts by considering every ball distinct, where balls of the same color are not distinct. Therefore, we must divide out those arrangements. There are $9!$ arrangements of the $9$ red balls that aren't distinct. Similarly, there are $7!$ and $3!$ ways to arrange for the other colors, so the answer is

$$\dfrac{19!}{9!7!3!} = 11085360$$