Say that a bag contains 100 marbles: 30 red, 30 blue, 30 green, plus a mix of 10 yellow and orange marbles. To be certain that you have 10 marbles of the same color, what is the minimum number you would need to remove (without looking) from the bag?
Since there are 3 colors with more than 10 marbles, you'd need 3*9+1 = 28 marbles to ensure that 10 are the same color (red,blue,green). Add to that the other 10 which are just nuisance marbles (since you can't ever get 10 orange or yellows), and you have 38 marbles.
Once you have drawn 27, you might have 9 each of red,blue,green. If you then draw all the yellows and oranges, you have taken 37 marbles. The next marble must be red,green, or blue.
To determine the minimum number of marbles you would need to remove (without looking) from the bag to be certain that you have 10 marbles of the same color, let's analyze the worst-case scenario.
The worst-case scenario would be if you first remove all the 100 non-yellow/non-orange marbles from the bag. In that case, you would have removed all the red, blue, and green marbles, leaving only the 10 yellow and orange marbles in the bag.
Now, you need to consider that the remaining 10 marbles could be a mix of yellow and orange marbles. To ensure you have at least 10 marbles of the same color, you would need to remove an additional 9 marbles. This is because you could remove 5 yellow and 4 orange marbles, or vice versa, resulting in a total of 9 marbles.
Therefore, the minimum number of marbles you would need to remove (without looking) from the bag to be certain that you have 10 marbles of the same color is 100 (to remove all the non-yellow/non-orange marbles) + 9 (to ensure you have 10 marbles of the same color) = 109 marbles.