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?

To determine the correctness of the statements, we use the worst case scenario principle. Statement 1: To find the smallest number n such that we must have one full group of at least one color, we first calculate the maximum number of balls we can draw without completing any single color group. This happens if we draw one ball less than the total for every color. Calculations: Black: 13 (14 minus 1) Blue: 19 (20 minus 1) Green: 25 (26 minus 1) Yellow: 27 (28 minus 1) Red: 37 (38 minus 1) White: 53 (54 minus 1) Sum = 13 + 19 + 25 + 27 + 37 + 53 = 174. The very next ball drawn (the 175th ball) must complete one of these color groups. Thus, n is 175. Statement 1 is correct. Statement 2: To find the smallest number m such that we must have at least one ball of each color, we look at the worst case where we exhaust all balls of all colors except the one with the smallest quantity. The colors and their quantities are: White (54), Red (38), Yellow (28), Green (26), Blue (20), and Black (14). The worst case is drawing every single ball of the five largest groups before getting a single ball from the smallest group (Black). Sum of the five largest groups = 54 + 38 + 28 + 26 + 20 = 166. The very next ball drawn (the 167th ball) must be from the remaining group (Black), ensuring we have at least one ball of each color. Thus, m is 167. Statement 2 is correct. Since both statements are correct, the answer is C.