A sample of 4 balls is to be selected at random from an urn containing 18 balls numbered 1 to 18. Five balls are green, 6 balls are white, and 7 balls are black.
(a) How many different samples can be selected?
samples
(b) How many samples can be selected that contain at least 1 white ball?
samples
(a) 18C4
(b) 18C4 - 12C4
To answer both questions, we can use the concept of combinations.
(a) To calculate the number of different samples that can be selected without any restrictions, we need to find the number of combinations. The formula for combinations is:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of items and r is the number of items being chosen without replacement.
In this case, we have 18 balls and we need to choose 4 of them. So the number of different samples that can be selected is:
C(18, 4) = 18! / (4!(18-4)!) = 18! / (4!14!) = 3060
Therefore, there are 3060 different samples that can be selected from the urn of 18 balls.
(b) To calculate the number of samples that contain at least 1 white ball, we can use the concept of complementary events. We calculate the total number of samples without any restrictions (as calculated in part a), and then subtract the number of samples that have no white balls. The formula for calculating the number of samples without white balls is:
C(13, 4) = 13! / (4!(13-4)!) = 13! / (4!9!) = 715
Therefore, the number of samples that contain at least 1 white ball is:
Total samples - Samples without white balls = 3060 - 715 = 2345
Hence, there are 2345 samples that can be selected from the urn containing at least 1 white ball.