First determine the greatest value of V such that there are NOT 14 balls of the same color.

If we have all 13 yellow balls, 13 green balls, 13 red balls, all 10 black balls, all 7 brown balls and 13 white balls, then we have a total of 69 balls and we DON'T have 14 balls of the same color.
At this point, there are 14 red balls, and 1 white balls remaining.
If we choose any of those balls, then we are guaranteed to have 14 balls of the same color.

In other words, if we have 70 balls, we are guaranteed to have 14 balls of the same color.

Answer: C