Please show your work. 1. An urn contains ten white balls numbered from 1 to 10,
ID: 3147304 • Letter: P
Question
Please show your work. 1. An urn contains ten white balls numbered from 1 to 10, and ten black balls numbered from 1 to 10. A sample of 5 balls is chosen from the urn. (a) How many different samples are there? (b) How many samples in (a) have at least one white ball? (c) How many samples in (a) have the property that the sum of the numbers on the balls is even? (d) How many samples in (a) have the property that the product of the numbers on the balls is even? (e) How many samples in (a) have the property that the numbers on the balls are distinct?Explanation / Answer
(a) Since there are 10 + 10 = 20 balls, we can choose 5 balls in 20C5 ways.
(b) Let us find the number of ways we can choose only black balls. Since there are 10 black balls and we need to choose 5, this can be done in 10C5 ways.
=> Number of ways we can choose 5 balls such that there is atleast one white ball = 20C5 - 10C5.
(c) The sum on the balls are even if there are even number of odd numbered balls.
This is half the total number or 20C5 / 2.
(d) Product of the numbers are even if there is atleast one even numbered ball.
If all balls are odd, then we can choose the 5 balls out of 10 odd numbered balls in 10C5 ways.
=> Number of samples where the product is even = 20C5 - 10C5.
(e) For each number, we can choose only one color - black or white. This can be done in 2 ways.
The 10 numbers can be chosen in 210 ways.