Jamestown Builders has a development of new homes. There are 6 different floor plans, 7 exterior colors, and an option of either a one-car or a two-car garage. How many choices are there for one home?
There are 6 floor plans, 7 exterior colors, and 2 garage options, so there are:
6 x 7 x 2 = 84 choices for one home.
If no digit may be used more than once, how many 5-digit numbers can be formed using only the digits 3, 8, 1, 2, 5, and 7?
To form a 5-digit number, we need to choose 5 digits from the given set of 6 digits without repeating any digit. This is a combination problem.
The number of ways to choose 5 digits out of 6 is given by:
6 choose 5 = 6!/[5!(6-5)!] = 6
Once we've chosen the 5 digits, we can arrange them in 5! = 120 ways.
So, the total number of 5-digit numbers that can be formed using only the digits 3, 8, 1, 2, 5, and 7 without repeating any digit is:
6 x 5! = 6 x 120 = 720.
Therefore, there are 720 such 5-digit numbers.
A drawer contains 2 red socks, 3 white socks, and 3 blue socks. Without looking, you draw out a sock, return it, and draw out a second sock. What is the probability that the first sock is blue and the second sock is red?
The probability of drawing a blue sock on the first draw is 3/8 because there are 3 blue socks out of a total of 8 socks. Since the sock is returned to the drawer, the probability of drawing a red sock on the second draw is 2/8 or 1/4 because there are now only 2 red socks out of 8 socks left in the drawer.
The probability of both events happening together is the product of their individual probabilities. So the probability of drawing a blue sock on the first draw and a red sock on the second draw is:
(3/8)*(1/4) = 3/32
Therefore, the probability of drawing a blue sock on the first draw and a red sock on the second draw is 3/32.
Two urns both contain green balls and red balls. Urn I contains 6 green balls and 4 red balls and Urn II contains 8 green balls and 7 red balls. A ball is drawn from each urn. What is the probability that both balls are red?
The probability of drawing a red ball from Urn I is 4/10 or 2/5, and the probability of drawing a red ball from Urn II is 7/15. Since the draws are independent, the probability of both events happening together is the product of their individual probabilities. So the probability of drawing a red ball from Urn I and Urn II is:
(2/5)*(7/15) = 14/75
Therefore, the probability of drawing a red ball from both urns is 14/75.