I think conceptually 2nd example is incorrect because in the 1st example each row was considered as independent entity to pick from.

lets say there r 2 lines (each line has thousands of people) and we need to pick one person from either of the line. How many ways can we do that? i would say 2C1. Pick 1 from 2 lines it doesn't matter how many are there in the lines.

so it should be 3C1 * 3C1 * 3C1 * ( 3C1 -- 1 object from any of the 3 rows).

I may be wrong

The right answer is 81. So option 1 is correct. I want to know how to get the factor of 2 in option 2 to reduce to option 1.

in the 2nd option it should not be 6c1 but it should be 3c1. That gives 81. Why 3c1? because we have to pick 1 from any of the 3 rows not 6. sorry i couldn't explain it any better.

Well you have to divide 6C1 in the option2 with 2 because there are 3 pairs of indistinguishable objects. (2 belonging two same row), hence the selections possible will be half.