This does not seem to be a typical "circular arrangement" question. The author just wants us to find out the number of arrangements where empty seats are not adjacent to each other.

The 3 empty seats are to be considered "identical" (Regardless of which of the 3 nominees sat on them prior to their getting empty). This is an important fact IMO. Now, out of 10 seats, 3 empty seats can be chosen in 10 C 3 ways, i.e., there is a total of 10 C 3 arrangements of 3 empty seats among 10 total seats. This part is easy to understand, since it is no different than a linear arrangement.

If you arrange adjacent numbers on a circle in pairs, you will come up with 10 pairs. This implies, a pair of empty seats could be arranged in 10 ways around a Round table. Now we are left with 1 empty seat to account for, and this could be 1 among any of the left 8 seats. Therefore, the total arrangements of a pair of empty seats are: 10 C 1 * 8 C 1.

Now, it gets little tricky. E 1, E 2, E 3 are the 3 empty seats and they are identical. All 3 empty seats could also appear adjacent to each other and they could be viewed as a pair of empty seats and an adjacent empty seat e.g., E 1 & E 2 could be a pair and E 3 could be the adjacent empty seat or E 2 & E 3 could be a pair and E 1 could be the adjacent empty seat. This implies that for each triplet there are 2 identical combinations ( Triplet means, 3 empty seats adjacent to each other). Therefore 1/2 of these should be subtracted from the total or else, they will be counted twice in the final tally. So, we have to find out how many such triplets are there.

Again, visualize a circle with 10 numbers. If you arrange them in triplets, you will find there are 10 triplets. Hence there will be 10 * 2 = 20 duplicate combinations. Therefore, we will deduct 10 from the total. The final Mathematical equation should look like this:

10 C 3 - {(10 C 1 * 8 C 1) - 10} / 10 C 3 = 5/12.

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