A set of five marbles is selected (without replacement) from a bag that contains 4 blue marbles, 3 green marbles, 6 yellow marbles, and 7 white marbles. How many sets of five marbles contain no more than two yellow ones?
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A bag contains 3 blue marbles, 9 green marbles, and 11 yellow marbles. Twice you draw a marble and replace it. Find P(blue, then green).
start fraction 27 over 529 end fraction
start fraction 27 over 23 end fraction
start fraction 15 over 529 end fraction
Start Fraction 12 over 23 End Fraction
To find the number of sets of five marbles that contain no more than two yellow ones, we will consider the different cases separately.
Case 1: No yellow marbles:
In this case, we need to select 5 marbles from the remaining 4 blue, 3 green, and 7 white marbles. So, the number of sets is given by:
(4 blue marbles) * (3 green marbles) * (7 white marbles) = 84
Case 2: One yellow marble:
In this case, we need to select 4 marbles from the remaining 4 blue, 3 green, 6 yellow, and 7 white marbles. Since we have one fixed yellow marble, we can select the remaining 4 marbles from the remaining 3 blue, 3 green, 6 yellow, and 7 white marbles. Therefore, the number of sets is given by:
(4 blue marbles) * (3 green marbles) * (6 yellow marbles) * (7 white marbles) * (choose 4 from 16 remaining marbles) = 3024
Case 3: Two yellow marbles:
In this case, we need to select 3 marbles from the remaining 4 blue, 3 green, 5 yellow, and 7 white marbles. Since we have two fixed yellow marbles, we can select the remaining 3 marbles from the remaining 3 blue, 3 green, 5 yellow, and 7 white marbles. Therefore, the number of sets is given by:
(4 blue marbles) * (3 green marbles) * (choose 3 from 15 remaining marbles) = 420
Now, add up the results from all the cases to get the total number of sets of five marbles that contain no more than two yellow ones:
84 + 3024 + 420 = 3528
Therefore, there are a total of 3,528 sets of five marbles that contain no more than two yellow ones.
To find the number of sets of five marbles that contain no more than two yellow ones, we can calculate the combinations of marbles that can be selected. Let's break down the problem step by step:
1. Calculate the number of ways to choose 0, 1, or 2 yellow marbles:
- For selecting 0 yellow marbles: There are 6 yellow marbles in the bag, and we need to choose none of them. This can be calculated as C(6, 0).
- For selecting 1 yellow marble: There are 6 yellow marbles, and we need to choose exactly one. This can be calculated as C(6, 1).
- For selecting 2 yellow marbles: There are 6 yellow marbles, and we need to choose exactly two. This can be calculated as C(6, 2).
2. Calculate the number of ways to choose the remaining marbles (non-yellow marbles):
- For selecting the remaining marbles, we need to consider the blue, green, and white marbles in the bag (since we have already considered the yellow marbles). We need to choose 5 marbles from a total of 4 blue marbles, 3 green marbles, and 7 white marbles. This can be calculated as C(4+3+7, 5).
3. Calculate the total number of sets:
- To find the total number of sets, we need to add the results from Step 1 and Step 2. This is because we can choose any combination of 0, 1, or 2 yellow marbles, and then choose the remaining marbles.
4. Calculate the final answer:
- To find the final answer, we add the results from Step 1 and Step 2: C(6, 0) x C(4+3+7, 5) + C(6, 1) x C(4+3+7, 5) + C(6, 2) x C(4+3+7, 5).
Now, let's calculate the answer:
C(6, 0) = 1
C(6, 1) = 6
C(6, 2) = 15
C(4+3+7, 5) = C(14, 5) = 2002
Final answer: 1 x 2002 + 6 x 2002 + 15 x 2002 = 22,066
Therefore, there are 22,066 sets of five marbles that contain no more than two yellow ones.