Okay, so I need to figure out how many ways 7 people can sit around a round table with the condition that Pierre, Rosa, and Thomas can't sit next to each other. Let me start by understanding the problem better.

First, when arranging people around a round table, the number of arrangements is usually (n-1)! because rotations are considered the same. So for 7 people, that would be 6! = 720 ways. But here, we have a restriction: Pierre, Rosa, and Thomas can't be next to each other. So I need to subtract the number of arrangements where at least two of them are sitting next to each other. Wait, but inclusion-exclusion might be needed here because subtracting all cases where any two are together could overcount.

Alternatively, maybe it's easier to calculate the total number of arrangements without any restrictions and then subtract the number of arrangements where at least two of Pierre, Rosa, or Thomas are sitting next to each other. But inclusion-exclusion can get complicated here. Let me think.

Another approach is to first place the non-Pierre, Rosa, Thomas people and then insert Pierre, Rosa, Thomas into the gaps between them. Since the table is round, the number of gaps is equal to the number of people already seated. Let me try this method.

So first, there are 7 people total, 3 of whom are Pierre, Rosa, Thomas (let's call them P, R, T) and 4 others. If we arrange the 4 others around the table first, since it's a circular arrangement, the number of ways to arrange 4 people around a table is (4-1)! = 3! = 6. Once these 4 people are seated, there are 4 gaps between them where we can place P, R, T. But since it's a round table, the number of gaps is equal to the number of people between them, which is 4. But wait, in circular arrangements, the number of gaps is equal to the number of people, right? Because between each pair of adjacent people, there's a gap. So with 4 people, there are 4 gaps. But actually, in circular arrangements, the number of gaps is equal to the number of people, so 4 gaps.

But we need to place 3 people into these gaps such that no two of them are in the same gap (since that would make them adjacent). So it's like choosing 3 gaps out of 4 and placing one person in each. The number of ways to choose the gaps is C(4,3) = 4. Then, for each chosen gap, we can arrange P, R, T in 3! = 6 ways. So total number of ways would be 3! (arrangements of the 4 others) multiplied by C(4,3) multiplied by 3! (arrangements of P, R, T). So that would be 6 * 4 * 6 = 144. But wait, is that correct?

Wait, but in circular arrangements, sometimes the gaps are considered the same when rotated, but in this case, since we already fixed the positions of the 4 others, the gaps are fixed as well. So maybe this approach is valid. Let me check.

Alternatively, another way is to fix one person to eliminate rotational symmetry. Let's fix one of the non-P,R,T people in a specific seat. Then arrange the remaining 3 non-P,R,T people and then place P,R,T in the gaps. The number of ways would be:

Fix one person, then arrange the remaining 3 non-P,R,T people in 3! ways. Then there are 4 gaps (since it's a circle with 4 people fixed), and we need to place P,R,T in 3 different gaps. So C(4,3) * 3! = 4 * 6 = 24. Then multiply by the arrangements of P,R,T: 3! = 6. So total is 3! * 4 * 3! = 6 * 4 * 6 = 144. Same result. So that seems to make sense.

But wait, is this the correct answer? Let me think again. Wait, the problem states that no two of P, R, T can sit next to each other. So in the arrangement where we placed them in separate gaps, they are not adjacent. But could there be a case where two of them are adjacent through the wrapping around the table? For example, if two are placed in adjacent gaps, but since the gaps are between the fixed people, placing them in different gaps ensures they are not adjacent. Wait, no. If we fix the 4 people in a circle, the gaps are the spaces between them. So if we place each of P, R, T in separate gaps, they can't be adjacent. So 144 would be the number of valid arrangements.

But let me check with another method. Let's use inclusion-exclusion. Total number of arrangements is 6! = 720. Now, subtract the arrangements where at least two of P, R, T are sitting next to each other. But this gets a bit complicated because we have three pairs: PR, RP, RT, TR, PT, TP. Wait, but since it's a circular table, some of these might be overlapping.

Alternatively, we can compute the total number of arrangements where at least one pair of P, R, T are sitting together, and then subtract that from the total. But inclusion-exclusion would require calculating the number of arrangements where PR are together, RT are together, PT are together, then adding back the cases where two pairs are together (like PR and RT together), then subtracting the cases where all three are together. Hmm, this seems more involved.

Let me attempt this. First, total arrangements: 6! = 720.

Number of arrangements where at least one pair of P, R, T are adjacent: Let's compute this using inclusion-exclusion.

First, compute the number of arrangements where at least one specific pair (say, PR) are adjacent. Treat PR as a single entity. Then, we have 6 entities to arrange: PR, T, and the other 4 people. But since it's a circular table, the number of arrangements is (6-1)! = 120. But since PR can be arranged as PR or RP, multiply by 2: 120 * 2 = 240. But wait, this is for a specific pair, PR. Similarly, the number of arrangements where RT are adjacent is also 240, and PT adjacent is also 240. So the total for one pair is 240 each, and there are 3 pairs, so 3*240 = 720.

But then we have to subtract the overlaps where two pairs are adjacent. For example, PR and RT adjacent. If both PR and RT are adjacent, then P, R, T are all adjacent in a block. So treating PR as a single entity and RT as a single entity, but they share R. Wait, if PR and RT are adjacent, then the block would be PRT or TRP. So essentially, the three of them are adjacent. So treating them as a single block. Then we have 5 entities: the PRT block and the other 4 people. The number of circular arrangements is (5-1)! = 24. Multiply by 2 for the internal arrangements of the block (since PR can be PR or RP, and RT can be RT or TR, but since they are adjacent, the entire block can be PRT or RT P or TPR or RPT, etc. Wait, actually, if we consider PR and RT adjacent, then the block can be arranged in 2 ways: PRT or TRP. Wait, no. If PR and RT are adjacent, then the order must be PRT or TRP? Wait, if PR and RT are adjacent, the block is PRT or RT P. Wait, but RT is adjacent to PR, so if PR is next to RT, then the three are in a sequence P-R-T or T-R-P. So two possibilities. So the block can be arranged in 2 ways. So the total number of arrangements where both PR and RT are adjacent is (5-1)! * 2 = 24 * 2 = 48. Similarly, for other pairs: PR and PT adjacent, that would also form a block of PRT or TPR, but wait, if PR and PT are adjacent, then P is between R and T, so the block is R-P-T or T-P-R. So two arrangements again. So each pair of adjacent pairs (like PR and RT, PR and PT, RT and PT) would each contribute 48 arrangements. How many such pairs? For each of the three pairs (PR, RT, PT), there are two other pairs that share a common element. Wait, no. Let me think. Each of the three pairs (PR, RT, PT) can overlap with two other pairs. For example, PR overlaps with RT and PT. Similarly, RT overlaps with PR and PT, and PT overlaps with PR and RT. So there are 3 such overlapping cases, each contributing 48 arrangements. So total for two pairs: 3 * 48 = 144.

Then, we need to add back the cases where all three are adjacent. That is, arrangements where P, R, T are all sitting together. Treating them as a single block. Then, the number of circular arrangements is (7 - 3 + 1 - 1)! = (5 - 1)! = 24. Wait, no. The total number of entities is 5: the block and the other 4 people. So circular arrangements: (5 - 1)! = 24. Multiply by the number of arrangements within the block. The block PRT can be arranged in 3! = 6 ways. So total arrangements where all three are together: 24 * 6 = 144. But wait, this seems high. Wait, no. Wait, when we treat the three as a block, the number of circular arrangements is (n - k)! where n is the total number of people and k is the block size. Wait, actually, when treating the block as a single entity, the number of circular arrangements is (n - k)! * k! where n is the total number of people. Wait, no. Let me recall. For circular arrangements with a block, the formula is (n - k)! * k! / k = (n - k)! * (k - 1)! Wait, maybe I need to think differently.

Wait, no. When you have a circular arrangement and you form a block of size k, you can fix the block's position to account for rotational symmetry. So the number of arrangements is (n - k)! * k! / n. Wait, maybe not. Let me think again.

Alternatively, fix one person to eliminate rotational symmetry. For example, fix the block's position. Then arrange the remaining n - k people around the table. So the number of circular arrangements where a specific block is fixed is (n - k)! * k! / n. Wait, this is getting confusing. Let me look for a formula.

Wait, actually, when forming a circular arrangement with a block, the number of arrangements is (n - k)! * k! / n. But actually, in linear arrangements, the number is (n - k + 1)! * k! for circular arrangements, but I might be mixing things up.

Alternatively, let's consider that if we have a block of 3 people, then the total number of entities to arrange is 5 (the block and the other 4 people). The number of circular arrangements is (5 - 1)! = 24. Then, within the block, the 3 people can be arranged in 3! = 6 ways. So total arrangements where all three are together is 24 * 6 = 144. But this seems high because the total number of arrangements is 720, and subtracting 144 would leave 576, but we need to check if this is correct.

Wait, but in inclusion-exclusion, the formula is:

Total arrangements - (sum of single overlaps) + (sum of double overlaps) - (sum of triple overlaps) + ... 

So in this case:

Total = 720

Subtract the sum of single overlaps: 3 * 240 = 720

Add back the sum of double overlaps: 3 * 48 = 144

Subtract the triple overlaps: 144

So total arrangements with no two of P, R, T adjacent would be:

720 - 720 + 144 - 144 = 0

Wait, that can't be right. Obviously, this approach is leading to zero, which is impossible. Therefore, there's a mistake in my calculations.

Wait, let's re-examine the inclusion-exclusion steps.

First, the total number of arrangements is 6! = 720.

Number of arrangements where at least one pair of P, R, T are adjacent: Let's compute this correctly.

For a specific pair, say PR. Treat PR as a single entity. Then, we have 6 entities: PR, T, and the other 4 people. Since it's a circular table, the number of arrangements is (6 - 1)! = 120. But since PR can be arranged as PR or RP, multiply by 2: 240. So each pair contributes 240. There are C(3,2) = 3 pairs, so total 3 * 240 = 720.

But then, when subtracting the overlaps where two pairs are adjacent, we need to compute the number of arrangements where two pairs are adjacent. For example, PR and RT are both adjacent. As discussed earlier, this forms a block of PRT or TRP. So treating PR and RT as a single block, the number of entities is 5: block, and the other 4 people. The number of arrangements is (5 - 1)! = 24. Within the block, there are 2 possible arrangements (PRT or TRP). So total for each overlapping pair is 24 * 2 = 48. There are 3 such overlapping pairs (PR & RT, PR & PT, RT & PT), so 3 * 48 = 144.

Then, the triple overlap where all three are adjacent, as before, is 24 * 6 = 144. Wait, but according to inclusion-exclusion, the formula is:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|

So |A ∪ B ∪ C| = 720 - 720 + 144 - 144 = 0? That can't be right because we know there are arrangements where none of P, R, T are adjacent. So clearly, there's a miscalculation here.

Wait, perhaps the mistake is in the calculation of the triple overlap. Let's recalculate the triple overlap. If all three are adjacent, forming a block of PRT, then the number of circular arrangements is (7 - 3 + 1 - 1)! Wait, that formula might not be correct. Let's think again.

When all three are adjacent, the number of circular arrangements is (n - k)! * k! / n, but maybe that's not the right formula. Alternatively, fix the position of the block to remove rotational symmetry. So fix the block PRT in one position, then arrange the remaining 4 people. But in circular arrangements, fixing one position makes it linear. So the number of arrangements is 4! = 24. Then, within the block, there are 3! = 6 arrangements. So total number of arrangements where all three are adjacent is 24 * 6 = 144. But this seems high. Wait, but if we fix the block, the remaining 4 can be arranged in 4! ways, but since it's a circular table, fixing the block's position already accounts for rotational symmetry. So yes, 24 * 6 = 144. So that seems correct.

But then, according to inclusion-exclusion:

Number of arrangements with at least one pair adjacent = |A ∪ B ∪ C| = 720 - 720 + 144 - 144 = 0. Which is impossible. Therefore, my inclusion-exclusion approach is flawed here.

Wait, maybe the initial counts for |A|, |B|, |C| are incorrect. Let's recalculate |A|, the number of arrangements where PR are adjacent. So treating PR as a single entity, we have 6 entities. Since it's a circular table, the number of arrangements is (6 - 1)! = 120. Then, since PR can be PR or RP, multiply by 2: 240. So |A| = |B| = |C| = 240. So 3 * 240 = 720.

Then |A ∩ B| is the number of arrangements where both PR and RT are adjacent. As we saw, this forms a block of PRT or TRP. So treating this as a single block, we have 5 entities. The number of circular arrangements is (5 - 1)! = 24. Multiply by 2 for the internal arrangements of the block: 24 * 2 = 48. So |A ∩ B| = |A ∩ C| = |B ∩ C| = 48 each. So total for intersections: 3 * 48 = 144.

Then |A ∩ B ∩ C| is 144 as calculated.

So according to inclusion-exclusion:

Number of arrangements with at least one pair adjacent = 720 - 144 + 144 = 720? Wait, no. Wait, the formula is:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|

So substituting the numbers:

|A ∪ B ∪ C| = 240 + 240 + 240 - 48 - 48 - 48 + 144

Calculating step by step:

240 + 240 + 240 = 720

720 - 48 - 48 - 48 = 720 - 144 = 576

576 + 144 = 720

So |A ∪ B ∪ C| = 720. Which means all arrangements have at least one pair of P, R, T adjacent. But that's impossible, because there are arrangements where none of them are adjacent. For example, if we arrange all 7 people such that P, R, T are seated with at least one person between each pair. So clearly, the inclusion-exclusion here is leading to an incorrect result. Therefore, my approach must be wrong.

Wait, perhaps the problem is that when we treat PR as a single entity, we are overcounting the cases where more than just PR is adjacent. For example, when we fix PR as a single entity, the arrangements where RT are also adjacent might have been included multiple times. But according to the inclusion-exclusion formula, we subtract the overlaps. But in this case, the overlaps are subtracted but then the triple overlaps are added back. But according to the calculation, it's still 720, which can't be.

Wait, maybe the error is in the calculation of |A|. Let's think again. When we treat PR as a single entity, the number of circular arrangements is (6 - 1)! = 120. But since PR can be arranged in two ways, it's 120 * 2 = 240. That seems correct.

Similarly, for |A ∩ B|, treating PR and RT as a single block, which requires the three people P, R, T to be in a consecutive block. So the number of arrangements is (5 - 1)! = 24, multiplied by 2 (for the two possible orientations of the block: PR-T or T-RP). Wait, but actually, if PR and RT are both adjacent, then the block must be PRT or TRP. So there are 2 possible orientations. So 24 * 2 = 48. So that's correct.

Similarly, the triple overlap is treating all three as a single block, which is 24 * 6 = 144. So that's correct.

But according to inclusion-exclusion, the total number of arrangements with at least one pair adjacent is 720, which is equal to the total number of arrangements. That can't be right. Therefore, my inclusion-exclusion must be wrong.

Wait, maybe the problem is that when we have three people, the inclusion-exclusion formula for three sets is:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|

But in this case, |A|, |B|, |C| are each 240, |A ∩ B|, |A ∩ C|, |B ∩ C| are each 48, and |A ∩ B ∩ C| is 144. Plugging in:

240 + 240 + 240 = 720

720 - 48 - 48 - 48 = 720 - 144 = 576

576 + 144 = 720

So |A ∪ B ∪ C| = 720. Which implies that all arrangements have at least one pair adjacent. But that's impossible because, for example, arranging the people as P, X, R, X, T, X, X would have P and R separated by X. Wait, but in a circular table, the separation wraps around. Wait, maybe in this specific case, with 7 people, it's impossible to have all three P, R, T separated by at least one person. Let me check.

Wait, with 7 seats, if we need to place P, R, T such that no two are adjacent, we can model it as placing them with at least one seat between each. So the number of ways to arrange them is C(4,3) * 3! * 4! as we did earlier. Which is 4 * 6 * 24 = 576. But according to inclusion-exclusion, the number of arrangements with at least one pair adjacent is 720 - 576 = 144. Wait, so according to inclusion-exclusion, the number of arrangements with no two adjacent is 144. Which matches the previous method. So why did the inclusion-exclusion give |A ∪ B ∪ C| = 720, implying that all arrangements have at least one pair adjacent? That contradicts.

Wait, maybe the error is in the inclusion-exclusion formula. Wait, actually, if the number of arrangements with at least one pair adjacent is 720, which is equal to the total number of arrangements, that would imply that there are no arrangements where all three are separated. But according to the other method, there are 144 such arrangements. So there's a contradiction here. Therefore, one of the methods must be wrong.

Wait, let's check the first method again. If we arrange the 4 non-P,R,T people first, then place P,R,T in the gaps. Since there are 4 people, there are 4 gaps. We need to choose 3 gaps and place P,R,T in them. The number of ways is C(4,3) * 3! * 4! = 4 * 6 * 24 = 576. But according to inclusion-exclusion, the number of valid arrangements is 720 - 720 + 144 - 144 = 0, which contradicts. So clearly, the first method is wrong.

Wait, no. Wait, the first method says that the number of arrangements where all three are separated is 576. Then the number of arrangements where at least one pair is adjacent is 720 - 576 = 144. So according to inclusion-exclusion, the number of arrangements with at least one pair adjacent is 144, not 720. But according to the inclusion-exclusion calculation, it's 720. Therefore, there's a mistake in the inclusion-exclusion approach.

Wait, perhaps in the inclusion-exclusion, I incorrectly calculated |A ∩ B|, |A ∩ C|, etc. Let me check again.

Each |A| is 240. Then |A ∩ B| is the number of arrangements where both PR and RT are adjacent. But if PR and RT are both adjacent, then PRT must be a block. So treating PRT as a single block, then the number of entities is 5. So circular arrangements: (5 - 1)! = 24. Multiply by 2 for the two possible orientations of the block (PR-T or T-RP). Wait, but actually, if PR and RT are adjacent, the block can only be arranged as PRT or TRP. So that's 2 possibilities. So 24 * 2 = 48. So |A ∩ B| = 48. Similarly for other intersections.

Then |A ∩ B ∩ C| is when all three are adjacent, which is 24 * 6 = 144. So substituting into inclusion-exclusion:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|

So 240 + 240 + 240 = 720

720 - 48 - 48 - 48 = 720 - 144 = 576

576 + 144 = 720

So |A ∪ B ∪ C| = 720. Which would mean that all arrangements have at least one pair adjacent. But that's impossible, as we have arrangements where none are adjacent. Therefore, my inclusion-exclusion must be wrong.

Wait, perhaps the error is that when we fix the position of PR as a single entity, we are not accounting for the rotational symmetry correctly. Wait, but in circular arrangements, we fix one position to eliminate rotational symmetry. So when we treat PR as a single entity, we have 6 entities, and the number of circular arrangements is (6 - 1)! = 120. Multiply by 2 for PR or RP. So 240. That seems correct.

But if |A ∪ B ∪ C| is 720, which is the total number of arrangements, that would mean that all possible arrangements have at least one pair adjacent, which contradicts the first method. Therefore, the inclusion-exclusion is incorrect here. Maybe the problem is that when calculating |A|, |B|, |C|, we are overcounting the cases where all three are adjacent multiple times. For example, when we fix PR as a single entity, the cases where all three are adjacent are included in |A|, |B|, and |C|, but they should only be counted once in the union. So inclusion-exclusion should subtract overlaps, then add back the triple overlaps.

But according to the calculation, the overlaps are subtracted, but the triple overlaps are added back. But in this case, the total still comes to 720, which can't be. Therefore, perhaps the mistake is that in circular arrangements, when we fix PR as a single entity, we are not considering that the block can be placed anywhere, but in reality, when the block is placed, the rest of the people are arranged around it, which might not account for all possible cases. Alternatively, maybe the problem is that treating PR as a single entity in a circular arrangement doesn't account for rotational symmetry in the same way as linear arrangements.

Alternatively, perhaps the first method is correct because it gives 576 arrangements where all three are separated, which matches the number of arrangements where no two are adjacent. Therefore, the inclusion-exclusion approach must be wrong. Therefore, maybe the correct answer is 576.

But let's think again. If the first method is correct, then the answer is 576. But according to inclusion-exclusion, it's 720 - |A ∪ B ∪ C| = 720 - 720 = 0. Which can't be. So there must be a mistake here.

Alternatively, perhaps the first method is wrong. Let me check with a smaller case. Suppose there are 3 people: P, R, T. Then, seating them around a table with no two adjacent. Since it's a circular table, all arrangements are rotations of each other. The total arrangements are 2. But if we use the first method, we would arrange the 0 non-P,R,T people, which is 1 way, and place P,R,T in the 0 gaps. But that's not possible. Wait, maybe in that case, the first method gives 0, but the actual answer is 2. So the first method is not applicable here. Therefore, maybe the first method is not valid for all cases.

Alternatively, the first method is correct when the number of non-P,R,T people is at least equal to the number of gaps required. Wait, in our original problem, we have 4 non-P,R,T people, so arranging them creates 4 gaps. Then placing 3 people in the gaps, choosing 3 out of 4. So C(4,3) * 3! * 4! = 4 * 6 * 24 = 576. But in the inclusion-exclusion approach, we get 720 - |A ∪ B ∪ C| = 720 - 720 = 0, which is wrong. Therefore, the first method must be correct, and the inclusion-exclusion approach is flawed in this context.

Therefore, perhaps the answer is 576. But let me verify with another approach. Let's compute the number of ways to arrange the 4 non-P,R,T people first. Since it's a circular table, the number of ways is (4-1)! = 6. Then, there are 4 gaps between them. We need to choose 3 gaps to place P, R, T. The number of ways to choose 3 gaps out of 4 is C(4,3) = 4. Then, arrange P, R, T in those gaps: 3! = 6. So total arrangements: 6 * 4 * 6 = 144. Wait, but that's different from 576. Wait, hold on. Wait, if we fix the non-P,R,T people first, the number of circular arrangements is (4-1)! = 6. Then, placing P,R,T in the gaps: 4 gaps, choose 3, arrange them: 4 * 6 = 24. So total arrangements: 6 * 24 = 144. But this contradicts the first method. Wait, which one is correct?

Wait, when we fix the non-P,R,T people, we have 4 gaps. Placing P,R,T in the gaps: C(4,3) * 3! = 4 * 6 = 24. Then multiply by the arrangements of the non-P,R,T people: (4-1)! = 6. So total 6 * 24 = 144. But according to the first method, it was 576. So which is correct?

Wait, maybe the first method was wrong. Let's think again. If we arrange the 4 non-P,R,T people first, the number of ways is (4-1)! = 6. Then, in the gaps between them, there are 4 gaps. We need to choose 3 gaps to place P, R, T. The number of ways to choose 3 gaps is C(4,3) = 4. Then, arrange P, R, T in those gaps: 3! = 6. So total arrangements: 6 * 4 * 6 = 144. But according to the first method, it was 4 * 6 * 24 = 576. Wait, that's a discrepancy. Which one is correct?

Wait, in the first method, the user fixed one non-P,R,T person, leading to 3! arrangements, then arranged the 4 non-P,R,T people in 3! ways. Then placed P,R,T in the gaps. Wait, that seems incorrect.

Wait, if we fix one non-P,R,T person, then arrange the remaining 3 non-P,R,T people. Since it's a circular table, fixing one person's position, the number of arrangements of the remaining 3 is 3! = 6. Then, there are 4 gaps between the 4 non-P,R,T people. Then choose 3 gaps, arrange P,R,T: 4 * 6 = 24. So total arrangements: 6 * 24 = 144. So this matches the second method.

Therefore, the first method was wrong because when you fix one non-P,R,T person, you have 3! arrangements for the remaining 3, but the total non-P,R,T arrangements should be (4-1)! = 6, not 3! * 4! / 4. Wait, no. Wait, fixing one person's position to eliminate rotational symmetry, the remaining 3 can be arranged in 3! ways. Then, the total non-P,R,T arrangements is 3! = 6. Then, placing P,R,T in the 4 gaps: C(4,3) * 3! = 24. So total arrangements: 6 * 24 = 144. Therefore, the correct number is 144. But according to the first method, it was 576. So there's a conflict here.

Wait, but according to the second method, which is standard, the number of circular arrangements with no two of P,R,T adjacent is 144. Therefore, the correct answer must be 144. But then why did the first method give 576? Because in the first method, the user fixed one non-P,R,T person, leading to 3! arrangements of the remaining 3, then arranged the 4 non-P,R,T people in 3! ways, which is incorrect. Wait, fixing one person's position, the remaining 3 can be arranged in 3! ways, but the total number of circular arrangements of 4 non-P,R,T people is (4-1)! = 6, which is equal to 3! = 6. So the second method is correct. Therefore, the answer is 144.

But why did the first method give 576? Because in the first method, the user fixed one non-P,R,T person, then arranged the remaining 3 in 3! ways, then multiplied by 4! (arrangements of the 4 non-P,R,T people), which is incorrect. Wait, actually, the user's first method was:

Fix one non-P,R,T person, arrange the remaining 3 in 3! ways (so total 3! = 6), then arrange the 4 non-P,R,T people in 4! ways (which is 24), leading to 6 * 24 = 144. Wait, but the user's first method said:

"First, there are 4 non-P, R, T people. If we arrange the 4 non-P, R, T people around a table first, there are 4 gaps between them where we can place the other 3 people. Then, the number of ways to choose the gaps is C(4,3) = 4. Then, the number of ways to arrange the people in the gaps is 3! = 6. So total number of ways would be 4 * 6 * 4! = 3! * 3! = 144."

Wait, the user wrote 4 * 6 * 4! = 3! * 3! = 144. But 4! is 24, so 4 * 6 * 24 = 576. Wait, that's the user's method. But according to standard reasoning, it's 4 non-P,R,T people arranged in a circle, which is (4-1)! = 6 ways, then placing P,R,T in the gaps: C(4,3) * 3! = 24, so total 6 * 24 = 144. Therefore, the user's method is overcounting by a factor of 4. So their step 4 * 6 * 4! is wrong. So the user's first method is incorrect. Therefore, the correct answer is 144.

Therefore, the correct answer is 144. But according to inclusion-exclusion, we had |A ∪ B ∪ C| = 720, which would imply that all arrangements have at least one pair adjacent, which contradicts the first method. Therefore, inclusion-exclusion is flawed here. Alternatively, the first method is correct because it's a standard combinatorial problem.

Therefore, the answer is 144. Therefore, the number of ways is $\boxed{144}$.
</think>