A box contains 14 black balls, 20 blue balls, 26 green balls, 28 yellow balls, 38 red balls and 54 white balls. Consider the following statements:
1. The smallest number n such that any n balls drawn from the box randomly must contain one full group of at least one colour is 175.
2. The smallest number m such that any m balls drawn from the box randomly must contain at least one ball of each colour is 167.

Which of the above statements is/are correct?

Correct Answer: (C) Both 1 and 2

For Statement 1, the worst-case scenario to avoid getting a complete set of any single color is drawing one less than the full count of each color: $(14-1) + (20-1) + (26-1) + (28-1) + (38-1) + (54-1) = 13 + 19 + 25 + 27 + 37 + 53 = 174$ balls. Drawing one more ball (the 175th) guarantees completing at least one color (Statement 1 is correct). For Statement 2, to guarantee drawing at least one ball of each of the 6 colors, consider the worst case where all balls of the five most numerous colors are drawn first: $54 + 38 + 28 + 26 + 20 = 166$ balls. The 167th ball drawn must be from the remaining black color (Statement 2 is correct).