The product of three numbers will equal ‘0’ if at least
one of the numbers is zero. Another way is to substract a number of combinations when none of the numbers equal zero from a total number of combinations.
A total number of combinations of 3 balls out of 6 is 6C3=20. The number of combinations of all ones is 3C3=1. Thus, the number of combinations of none zero out of three is 20-1=19.
Double check the longer way (at least
Number of combinations of exactly one zero (two '1'): 3C1*3C2=9
Number of combinations of exactly two zeros (one '1'): 3C2*3C1=9
Number of combinations of exactly tree zeros: 3C3=1
Add them all up and the answer is 19.