**Solution:**

To solve this problem, we will use the concept of permutations and combinations. Let's break down the solution into smaller steps:

**Step 1: Total Arrangements Without Any Restrictions**

Initially, let's consider the total number of arrangements without any restrictions. Since there are 12 people, the total number of arrangements would be 12!.

**Step 2: Arrange the Two Particular Persons Side by Side**

Now, let's consider the two particular persons who should be side by side. We can treat them as a single entity or group. Therefore, we have 11 entities (10 remaining people + 1 group of 2 persons) to arrange.

The number of arrangements of these 11 entities is 11!.

**Step 3: Consider the Two Persons as One Entity**

Since the two particular persons should not be side by side, we need to subtract the number of arrangements where they are together from the total arrangements without any restrictions.

To calculate this, we treat the two persons as one entity or group. This group can be arranged in 2! (2 factorial) ways.

**Step 4: Calculate the Total Arrangements**

To calculate the total number of arrangements where the two particular persons are not side by side, we subtract the arrangements where they are together from the total arrangements without any restrictions.

Total arrangements = Total arrangements without restrictions - Arrangements where the two persons are together

Total arrangements = 12! - (11! * 2!)

Now, let's calculate the value:

12! = 479,001,600
11! = 39,916,800
2! = 2

Total arrangements = 479,001,600 - (39,916,800 * 2) = 479,001,600 - 79,833,600 = 399,168,000

Therefore, the total number of arrangements where the two particular persons are not side by side is 399,168,000.