The probability of picking up sesame dumplings, peanut dumplings and bean paste dumplings is respectively

6/(6+5+4)=2/5

5/(6+5+4)= 1/3

4/(6+5+4)=4/ 15.

The probability of each kind of glutinous rice balls going at least once should be (2/5) * (1/3) * (4/15) = 8/225.

I thought the same thing.