The question involves simple permutation of arranging 8 identical balls in two rows; one with 6 locations and another with 2 locations. The 6-ball row has 8 blocks while the 2-ball row only has 4 blocks.

Option 1: The calculation here is incorrect because 8 balls cannot be placed in the two rows in 38 unique ways.

Option 2: This is the correct answer. You can choose two locations for the two balls from the four available in the second row in 4C2=6 ways. Then, you can choose six locations for the remaining six balls from the eight available in the first row in 8C6=28 ways. Multiplying these together gives 6*28=168 ways. Since balls are identical, each arrangement appears 8!/(6!*2!) = 28 times. The total number of unique arrangements is thus 168/28 = 6 ways.

Option 3: Here the calculation seems incorrect because it does not follow the correct permutation principle for identical items

Option 4: Although it involves the number 2 (balls in the second row), this option is incorrect because it doesn`t correctly calculate the total permutations.