Can you help me understand this math problem...I could figure it out the long way but I think there is a formula that I can use that I'm not remembering:
I have 18 different colors of marbles. I'm going to arrange them in sets of three.
I want to do this 6 times with out any of them ever being in the same set again...like the first time I might have orange, purple, green in one set and so the next time, I don't want there to be a set with any of those three together.
If it doesn't work right with doing it six times, I think it should with 5 times.
Any help?
Yes, I can help you with that math problem.
To solve this problem, we can use the concept of combinations. We want to arrange 18 colors of marbles in sets of 3, without any repetition of sets in 6 rounds (or 5 rounds if necessary).
In the first round, we can select any 3 marbles from the 18 available. So, there are a total of 18 choose 3 ways to select the marbles for the first round, which is denoted as C(18, 3) or written as 18C3.
After selecting the marbles for the first round, there are 15 marbles left for the second round. However, since we want to avoid repeating any sets, we need to subtract the combinations that would include any of the marbles already chosen.
To calculate the number of combinations for the second round, we need to consider two cases:
Case 1: There is no overlap with the marbles from the first round.
In this case, we can choose any 3 marbles from the remaining 15. So, there are 15C3 combinations.
Case 2: There is overlap with one or more marbles from the first round.
In this case, we need to consider the number of combinations that include each overlap. Since we want to avoid using any of those combinations, we subtract them from the total combinations. The number of combinations with an overlap of i marbles is C(3, i) * C(15, 3 - i). We consider i marbles from the 3 chosen in the first round, and the remaining 3 - i marbles from the 15 remaining marbles.
We repeat this process for each round, adjusting the number of available marbles each time.
To find the total number of combinations for 6 rounds, we multiply the number of combinations in each round together:
Total combinations = C(18, 3) * (C(15, 3) - (C(3, 1) * C(15, 2)) - (C(3, 2) * C(15, 1)) - (C(3, 3) * C(15, 0)))
If you want to calculate the total combinations for 5 rounds, you would repeat the process but stop after the fifth round, using 13 marbles for the last round.
I hope this helps you understand the problem better. If you have any further questions, please let me know!
Yes, I can help you understand this math problem. You are trying to arrange 18 different colors of marbles into sets of three, and you want to make sure that none of the sets have any of the same three colors. You want this arrangement to be done either 6 or 5 times.
To solve this problem, we can use a concept called combinations. In mathematics, a combination is a selection of items from a larger set without regard to the order of the items.
First, let's calculate the total number of different sets you can make with 18 marbles taken 3 at a time. This can be done using the formula for combinations, which is given by:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of items, r is the number of items being selected (in this case, 3), and ! denotes the factorial operator.
Using this formula, we can calculate the number of different sets possible for 18 marbles taken 3 at a time:
C(18, 3) = 18! / (3!(18-3)!) = 18! / (3!15!) = (18 * 17 * 16) / (3 * 2 * 1) = 816
So, there are a total of 816 different sets you can make with the 18 marbles taken 3 at a time.
Now, let's consider the constraint that none of the sets should have the same three colors. If you want to arrange the marbles into sets 6 times, you'll need 6 sets, each containing 3 different marbles. Following this constraint, the total number of sets you can make is:
816 / 6 = 136
So, you can make 136 different sets by arranging the marbles into sets of 3, while ensuring that no three colors repeat over 6 sets.
If you want to do it 5 times instead, the number of sets you can make would be:
816 / 5 = 163.2
Since you can't have a fraction of a set, you would need to round down to the nearest whole number. Therefore, you can make 163 different sets over 5 iterations.
I hope this explanation helps you understand and solve the math problem!