Correct Option (3)
The total number of balls in the box is 14 (Black) + 20 (Blue) + 26 (Green) + 28 (Yellow) + 38 (Red) + 54 (White) = 180 balls.
Statement 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.
To determine the smallest number 'n' that guarantees a full group of at least one colour, we consider the worst-case scenario. This scenario involves drawing the maximum possible number of balls without completing any single colour group. This occurs when we draw one ball less than the total count for each of the six colours.
- Number of balls drawn in the worst-case = (14-1) + (20-1) + (26-1) + (28-1) + (38-1) + (54-1)
- = 13 + 19 + 25 + 27 + 37 + 53 = 174 balls.
After drawing 174 balls, it is possible that exactly one ball of each colour remains in the box, meaning no full group has been formed. The next ball drawn (the 175th ball) must complete one of the existing colour groups. Therefore, n = 174 + 1 = 175. Statement 1 is correct.
Statement 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.
To determine the smallest number 'm' that guarantees at least one ball of each colour, we again consider the worst-case scenario. This involves drawing all balls of the colours that are not the least numerous, before drawing any ball of the least numerous colour. The colour with the fewest balls is Black, with 14 balls.
- Number of balls drawn in the worst-case = Total balls - (Number of balls of the least numerous colour)
- = 180 - 14 (Black balls) = 166 balls.
After drawing 166 balls, it is possible that all these balls belong to the Blue, Green, Yellow, Red, and White categories, and no Black ball has been drawn. The next ball drawn (the 167th ball) must be a Black ball, thereby ensuring that at least one ball of each colour is present among the drawn balls. Therefore, m = 166 + 1 = 167. Statement 2 is correct.
Since both Statement 1 and Statement 2 are correct, option (3) is the correct answer.
Incorrect Options:
Options (1), (2), and (4) are incorrect because both Statement 1 and Statement 2 have been determined to be factually correct based on the principles of the Pigeonhole Principle applied to probability scenarios.