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 ?

• Total balls = 14 + 20 + 26 + 28 + 38 + 54 = 180.

• Statement 1: To guarantee a full group of one color, take the maximum number of balls without completing any group: (14-1) + (20-1) + (26-1) + (28-1) + (38-1) + (54-1) = 13 + 19 + 25 + 27 + 37 + 53 = 174.

• Thus, the smallest n is 174 + 1 = 175, making Statement 1 correct.

• Statement 2: To guarantee at least one ball of each color, take the maximum number of balls while missing at least one color. The largest group is White (54), so exclude all 54 White balls and take all others: 180 - 54 = 126.

• Wait, the logic for 'at least one of each' is: Max balls without having all colors = Total - (Size of largest group). If we miss White, we have 126 balls. If we miss Red, we have 180-38=142. The maximum is 142 (missing Red). So m = 142 + 1 = 143? No, let's re-read carefully. 'Must contain at least one ball of each colour'. Worst case is missing the largest group. Max balls without all colors = Sum of all except the largest group = 180 - 54 = 126? No, if I pick 126 balls, I could have picked all non-white balls. But I could also pick 142 balls (all except Red). 142 > 126. So max balls without all colors is 142. So m = 143. Why is the key saying 167 is correct? Let me re-calculate. Ah, I see. The question asks for the smallest m such that ANY m balls drawn MUST contain at least one of each. This is equivalent to Total - (Max balls that can be drawn while missing at least one color). The max balls missing one color is the sum of the other 5 colors. The largest sum of 5 colors is obtained by excluding the smallest color. Smallest color is Black (14). Sum of others = 180 - 14 = 166. So if I draw 166 balls, I might have drawn all except Black. Therefore, to guarantee I have Black (and thus all colors), I need 166 + 1 = 167. Statement 2 is correct.