Explanation:

Correct option 3: Both 1 and 2

Given:

The box contains:

• Black = 14

• Blue = 20

• Green = 26

• Yellow = 28

• Red = 38

• White = 54

Total = 14 + 20 + 26 + 28 + 38 + 54 = 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.”

We are looking for the minimum number of balls such that no matter what, you end up with at least one full set of a single color (i.e., all balls of that color).

To avoid completing any full color, you should try to draw the maximum number of balls without taking all of any one color.

This is done by taking (total of each color - 1):

(14−1)+(20−1)+(26−1)+(28−1)+(38−1)+(54−1)=13+19+25+27+37+53=174

So, in the worst-case, you can draw 174 balls without completing any full group.

Thus, on drawing 175 balls, you must complete at least one group.

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 avoid getting at least one ball from each color, we consider the worst-case:
Take the maximum possible number of balls from only 5 colors, and none from the 6th.

We exclude the color with least number of balls (Black = 14), so take all balls from the other 5:

Blue(20)+Green(26)+Yellow(28)+Red(38)+White(54)=166

So, in worst-case, you can draw 166 balls and still miss Black.

Therefore, on the 167th draw, you must get the 6th color, completing at least one ball of each color.Statement 2 is also correct.

Final Answer:

Both statements 1 and 2 are correct.