A sample of 4 balls is to be selected at random from an urn containing 20 balls numbered 1 to 20. Six balls are green, 6 balls are white, and 8 balls are black.
(a) How many different samples can be selected?
(b) How many samples can be selected that contain at least 1 white ball?
i think A is 288, And im sorry but i dont know B.
Also i want to have A confirmed by a (Real) Teacher, or someone who DOES know this, so i dont end up giving you wrong info. Sorry for the lack of answer(s) I tried
To find the number of different samples that can be selected from the urn, we can use the concept of combinations.
(a) The total number of different samples can be obtained by calculating the number of combinations of 4 balls selected from a total of 20 balls. The formula to calculate combinations is given by:
nCr = n! / r!(n-r)!
where n is the total number of objects, r is the number of objects chosen, and "!" represents factorial.
For this problem, we need to calculate 20C4:
20C4 = 20! / (4!(20-4)!)
Simplifying the expression:
20C4 = 20! / (4! * 16!)
Now, let's calculate the factorial values using the formula:
20! = 20 × 19 × 18 × 17 × 16!
4! = 4 × 3 × 2 × 1
16! = 16 × 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
Plugging these values into the combination formula:
20C4 = (20 × 19 × 18 × 17 × 16!) / (4 × 3 × 2 × 1 × 16!)
Simplifying further:
20C4 = (20 × 19 × 18 × 17) / (4 × 3 × 2 × 1)
Calculating this expression gives us:
20C4 = 4845
So, there are 4845 different samples that can be selected from the urn.
(b) To calculate the number of samples that contain at least 1 white ball, we can subtract the number of samples without any white balls from the total number of samples.
The number of samples without any white balls can be calculated by finding the number of combinations of 4 balls selected from the 14 non-white balls (6 green + 8 black).
14C4 = 14! / (4!(14-4)!) = 14! / (4! × 10!)
Using the factorial formula, we get:
14! = 14 × 13 × 12 × 11 × 10!
Calculating further:
14C4 = (14 × 13 × 12 × 11 × 10!) / (4! × 10!)
Simplifying:
14C4 = (14 × 13 × 12 × 11) / (4 × 3 × 2 × 1)
Calculating this expression gives us:
14C4 = 1001
Therefore, the number of samples that contain at least 1 white ball is:
4845 - 1001 = 3844