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, we need to consider the different combinations of marbles that can be drawn from the bag.
First, let's determine the total number of ways to draw four marbles from the bag without any restrictions. We can use the concept of combinations, specifically the combination formula.
The combination formula is given as:
C(n, r) = n! / (r! * (n - r)!)
Where C(n, r) represents the number of combinations of n items taken r at a time, and n! represents the factorial of n.
In this case, we have a total of 12 marbles in the bag, and we want to draw 4 marbles. So we can calculate the total number of combinations as:
C(12, 4) = 12! / (4! * (12 - 4)!)
Next, let's find the number of ways to draw exactly two red marbles from the bag. We can calculate this by considering the combinations of selecting 2 red marbles out of the 4 available, which can be represented as C(4, 2).
Finally, to find the number of different samples that include exactly two red marbles, we can multiply the number of ways to draw exactly two red marbles (C(4, 2)) by the number of ways to draw the remaining two marbles from the remaining 10 marbles in the bag (C(10, 2)).
So the number of different samples that include exactly two red marbles can be calculated as:
C(4, 2) * C(10, 2)
Now we can plug the values into the formulas and calculate the result:
C(4, 2) = 4! / (2! * (4 - 2)!) = 6
C(10, 2) = 10! / (2! * (10 - 2)!) = 45
Therefore, the number of different samples that include exactly two red marbles is:
6 * 45 = 270
So there are 270 different samples possible which include exactly two red marbles.