A bag contains three red marbles, three green ones, one lavender one, five yellows, and two orange marbles.
How many sets of five marbles include either the lavender one or exactly one yellow one but not both colors?
To find the number of sets of five marbles that include either the lavender one or exactly one yellow one but not both colors, we need to count the number of ways we can select marbles from the given bag that meet these conditions.
First, let's count the number of sets of five marbles that include the lavender one. Since we are selecting five marbles in total, we have 4 more marbles to select along with the lavender one.
We can choose the remaining marbles from the red, green, yellow, or orange marbles. There are 3 red marbles, 3 green marbles, 5 yellow marbles, and 2 orange marbles in the bag.
Considering selecting from these remaining marbles, we have the following cases:
Case 1: Selecting 4 marbles from the remaining marbles (excluding the yellow marbles):
In this case, we have 3 options for each selection since we can choose from the 3 red or 3 green or 2 orange marbles. So the total number of ways to select 4 marbles from the remaining marbles is 3 + 3 + 2 = 8.
Case 2: Selecting 3 marbles from the remaining marbles:
In this case, we need to select 2 marbles from the red, green, or orange marbles. We have 3 options for each selection. So the total number of ways to select 3 marbles from the remaining marbles is 3 + 3 + 3 = 9.
Therefore, the total number of sets of five marbles that include the lavender one is 8 + 9 = 17.
Next, let's count the number of sets of five marbles that include exactly one yellow one. Since we already used the lavender marble for the previous case, we cannot include it in this case.
We can choose the remaining four marbles from the red, green, yellow (excluding the lavender one), or orange marbles. Therefore, we have the following cases:
Case 3: Selecting 4 marbles from the remaining marbles (excluding yellow and lavender):
In this case, we need to select all 4 marbles from the red or green or orange marbles. So the total number of ways to select 4 marbles from the remaining marbles is 3 + 3 + 2 = 8.
Therefore, the total number of sets of five marbles that include exactly one yellow one is 8.
Now, let's count the number of sets of five marbles that include either the lavender one or exactly one yellow one but not both colors.
By combining the results from cases 1, 2, and 3, we get:
Number of sets = Number of sets with lavender + Number of sets with exactly one yellow - Number of sets where both lavender and one yellow are present
Number of sets = 17 + 8 - 1 (since both lavender and one yellow option is counted twice in case 2)
Number of sets = 24
Therefore, the number of sets of five marbles that include either the lavender one or exactly one yellow one but not both colors is 24.
To find the number of sets of five marbles that include either the lavender one or exactly one yellow one but not both colors, we need to consider two different cases:
Case 1: Sets with the lavender marble:
In this case, we need to choose 4 more marbles from the remaining marbles (excluding the lavender marble and the yellow ones). The remaining marbles consist of 3 red, 3 green, 5 yellow, and 2 orange marbles. So, we have a total of (3+3+2) = 8 marbles to choose from. The number of ways to choose 4 marbles from these 8 is given by the combination formula:
C(8, 4) = 8! / (4! * (8-4)!) = 70
Case 2: Sets with exactly one yellow marble:
In this case, we need to choose 1 yellow marble and 4 more marbles from the remaining marbles (excluding the yellow marbles and the lavender one). The remaining marbles consist of 3 red, 3 green, and 2 orange marbles. So, we have a total of (3+3+2) = 8 marbles to choose from. The number of ways to choose 1 yellow marble from the 5 yellow marbles is given by the combination formula:
C(5, 1) = 5! / (1! * (5-1)!) = 5
Then, we need to choose 4 marbles from the remaining 8 marbles (excluding the chosen yellow marble and the lavender marble). The number of ways to choose 4 marbles from these 8 is given by the combination formula:
C(8, 4) = 8! / (4! * (8-4)!) = 70
Now, to find the total number of sets, we add the number of sets from each case:
Total number of sets = Number of sets with the lavender marble + Number of sets with exactly one yellow marble
= 70 + 5
= 75
Therefore, there are 75 sets of five marbles that include either the lavender one or exactly one yellow one but not both colors.
number with 1 Lavender = 1 x C(13,4) = 715
number with 1 yellow = C(5,1) x C(9,4) = 630
exclude yellow and lavender = C(8,5) = 56
number as described = 715+630-56 = 1289