A bag contains eight yellow marbles, seven red marbles, and five black marbles. Without looking in the bag, Igor removes $N$ marbles all at once. If he is to be sure that, no matter which choice of $N$ marbles he removes, there are at least four marbles of one colour and at least three marbles of another colour left in the bag, what is the maximum possible value of $N$ ?
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
Solution 1

Suppose that Igor has removed some balls from the bag, and the remaining balls do not satisfy the required condition. What is the maximum number of balls that can remain? In order to
not satisfy the required condition, either there are not 4 balls of any colour (so the maximum number is 9 balls, ie. 3 of each colour) or there are at least 4 balls of one colour, but there are not 3 of either of the other colours.

In this second case, we could have 2 balls of each of two colours, and as many as possible of the third colour. The maximum number of balls of any colour that can be in the bag is 8 (the number of yellow balls with which Igor starts). So the maximum number of balls still in the bag in this case is 12 .

Therefore, if Igor removes 8 or more balls, then the remaining balls might not satisfy the required condition.

However, if Igor removes 7 or fewer balls, then the remaining balls will satisfy the required condition, since the maximum number of balls in any case which does not satisfy the condition is 12 .

Therefore, the maximum possible value of $N$ is 7 .

## Solution 2

Since we want to determine the maximum possible value of $N$, we start with the largest of the answers and rule out answers until we come to the correct answer.

If Igor removed 10 marbles, he might remove 5 red and 5 black marbles, leaving 8 yellow, 2 red, and 0 black marbles, which does not meet the required condition.

Thus, 10 is not the answer.

If Igor removed 9 marbles, he might remove 5 red and 4 black marbles, leaving 8 yellow marbles, 2 red marbles, and 1 black marble, which does not meet the required condition.

Thus, 9 is not the answer.

If Igor removed 8 marbles, he might remove 5 red and 3 black marbles, leaving 8 yellow, 2 red, and 2 black marbles, which does not meet the required condition.

Thus, 8 is not the answer.

Is 7 the answer?

There are $8+7+5=20$ marbles to begin with. If 7 are removed, there are 13 marbles left.

Since there are 13 marbles left, then it is not possible to have 4 or fewer marbles of each of the three colours (otherwise there would be at most 12 marbles). Thus, there are at least 5 marbles of one colour.

Could there be 2 or fewer marbles of each of the other two colours? If so, then since there are 13 marbles in total, there must be at least 9 marbles of the first colour. But there cannot be 9 or more marbles of any colour, as there were at most 8 of each colour to begin with. Therefore, there must be at least 3 of one of the other two colours of marbles.

This tells us that if 7 marbles are removed, there are at least 5 marbles of one colour and 3 of another colour, so choosing $N=7$ marbles guarantees us the required condition.

Therefore, 7 is the maximum possible value of $N$.

ANSWER: (B)