Calculating the total number of arrangements:
To calculate the total number of ways 6 girls and 6 boys can be arranged around a table, we need to first consider the total number of people to be seated which is 12.
Total number of ways to arrange 12 people around a table = (12-1)! = 11!
So, 11! = 39,916,800

Calculating the number of arrangements when 2 boys are adjacent:
Now, let's calculate the number of ways when 2 boys are seated together. Consider the 2 boys as a single entity.
Total number of ways to arrange 11 people (including the pair of boys as one) around a table = (11-1)! = 10!
The number of ways to arrange the 2 boys among themselves = 2!
So, the total number of ways to arrange 2 boys and 10 other people = 10! * 2!
= 725,760

Calculating the number of arrangements when 2 boys are not adjacent:
To calculate the number of ways when 2 boys are not seated together, we need to subtract the number of ways when 2 boys are adjacent from the total number of ways to arrange 12 people.
Number of ways when 2 boys are not adjacent = Total number of ways - Number of ways when 2 boys are adjacent
= 39,916,800 - 725,760
= 39,191,040
Therefore, there are 39,191,040 different ways to arrange 6 girls and 6 boys around a table such that 2 boys are not adjacent.