A basket contains 15 red numbered balls and 10 white numbered balls. A sample of five balls is selected. How many samples contain three red balls and two white balls? Explain your answer.

you want to choose 3 of the 15 reds and 2 of the 10 whites

Number of ways = C(15,3) x C(10,2)
= 455(45) = 20475

To find out how many samples contain three red balls and two white balls, we need to use the concept of combinations.

First, let's determine the number of ways we can choose 3 red balls from the 15 red numbered balls. This is denoted as "15 choose 3" and can be calculated using the combination formula:

C(n, r) = n! / (r!(n-r)!),

where C(n, r) represents the combination of n items taken r at a time, and "!" denotes the factorial operation.

For this case, we have n = 15 (red balls) and r = 3 (red balls chosen). Plugging these values into the formula:

C(15, 3) = 15! / (3!(15-3)!) = 15! / (3!12!) = (15 x 14 x 13) / (3 x 2 x 1) = 455.

Therefore, there are 455 ways to select 3 red balls from the 15 available.

Now, let's determine the number of ways we can choose 2 white balls from the 10 white numbered balls. Following the same steps, we have:

C(10, 2) = 10! / (2!(10-2)!) = 10! / (2!8!) = (10 x 9) / (2 x 1) = 45.

There are 45 ways to select 2 white balls from the 10 available.

Since we want to find the number of samples that contain 3 red balls and 2 white balls, we need to multiply the two combinations we calculated:

Number of samples = C(15, 3) * C(10, 2) = 455 * 45 = 20,475.

Hence, there are 20,475 samples that contain three red balls and two white balls.