A bag contains 4 blue, 4 red, and 4 green marbles. Four marbles are drawn at random from the bag. How many different samples are possible which include exactly two red marbles?

To find the number of different samples that include exactly two red marbles out of the four drawn from the bag, we need to consider the following:

1. Selecting two red marbles: We have 4 red marbles in the bag, and we need to choose 2 of them. This can be done using the combination formula. The number of ways to choose r items from a set of n items is given by the formula: C(n, r) = n! / (r! * (n - r)!). In this case, we have 4 red marbles and we need to choose 2, so we have C(4, 2) = 4! / (2! * (4 - 2)!) = 6 ways to choose exactly two red marbles.

2. Selecting the remaining two marbles: After selecting two red marbles, we need to select two more marbles from the remaining 10 marbles in the bag (4 blue and 4 green). Again, we can use the combination formula to find the number of ways to choose 2 marbles from the remaining 10. Considering that the order of selection doesn't matter, we have C(10, 2) = 10! / (2! * (10 - 2)!) = 45 ways to choose the remaining two marbles.

3. Combining both selections: To find the total number of different samples that include exactly two red marbles, we multiply the number of ways to choose the red marbles (6) by the number of ways to choose the remaining marbles (45): 6 * 45 = 270.

Therefore, there are 270 different samples possible that include exactly two red marbles when four marbles are drawn from the bag.