A jar contains six red marbles, four green marbles and five yellow ones.
a) determine the number of ways in which any thee marbles can be selected?
b) Determine the number of ways to select four marbles where two are red and two are yellow?
c) Determine the number of ways in which six marbles can be selected where two are red two are green and two are yellow?
so for question a i came up with
p(15,3) = 455
for question b i was thinking
p(4,2) x p (4,2)
and for c
p(6,2) x p(6,2) x (p(6,2)
not sure if those are close though.
Let's break down each question and calculate the answers step by step.
a) To determine the number of ways three marbles can be selected from a jar, we can use the combination formula. The combination formula calculates the number of ways to select a certain number of items from a larger set, without considering their order.
The formula for combinations is given by:
nCr = n! / (r!(n-r)!)
Where n is the total number of items and r is the number of items to be chosen.
In this case, there are 6 red marbles, 4 green marbles, and 5 yellow marbles. We want to choose three marbles, so n = 15 and r = 3.
Plugging the values into the combination formula:
nCr = 15! / (3!(15-3)!)
= 15! / (3!12!)
= (15 * 14 * 13 * 12!) / (3! * 12!)
= (15 * 14 * 13) / (3 * 2 * 1)
= 455
Therefore, the number of ways to select any three marbles from the jar is 455.
b) To determine the number of ways to select four marbles where two are red and two are yellow, we need to calculate the combinations separately for red and yellow marbles.
For the combination of red marbles, we have 6 red marbles and we want to choose 2, so n = 6 and r = 2:
Combination of red marbles: nCr = 6! / (2!(6-2)!) = 6! / (2!4!) = (6 * 5) / (2 * 1) = 15
For the combination of yellow marbles, we have 5 yellow marbles and we want to choose 2, so n = 5 and r = 2:
Combination of yellow marbles: nCr = 5! / (2!(5-2)!) = 5! / (2!3!) = (5 * 4) / (2 * 1) = 10
To calculate the total number of ways to select four marbles with two red and two yellow, we multiply the combinations calculated for each color:
Total number of ways = Combination of red marbles * Combination of yellow marbles
= 15 * 10
= 150
Therefore, the number of ways to select four marbles with two red and two yellow is 150.
c) To determine the number of ways to select six marbles where two are red, two are green, and two are yellow, we will calculate the combinations for each color separately, and then multiply them together.
For the combination of red marbles, we have 6 red marbles and we want to choose 2, so n = 6 and r = 2:
Combination of red marbles: nCr = 6! / (2!(6-2)!) = 6! / (2!4!) = (6 * 5) / (2 * 1) = 15
For the combination of green marbles, we have 4 green marbles and we want to choose 2, so n = 4 and r = 2:
Combination of green marbles: nCr = 4! / (2!(4-2)!) = 4! / (2!2!) = (4 * 3) / (2 * 1) = 6
For the combination of yellow marbles, we have 5 yellow marbles and we want to choose 2, so n = 5 and r = 2:
Combination of yellow marbles: nCr = 5! / (2!(5-2)!) = 5! / (2!3!) = (5 * 4) / (2 * 1) = 10
To calculate the total number of ways to select six marbles with two red, two green, and two yellow, we multiply the combinations calculated for each color:
Total number of ways = Combination of red marbles * Combination of green marbles * Combination of yellow marbles
= 15 * 6 * 10
= 900
Therefore, the number of ways to select six marbles with two red, two green, and two yellow is 900.