Answer:
1,968
Step-by-step explanation:
Let x₁ and x₂, y₁ and y₂, and z₁ and z₂ represent the 3 pairs of siblings, and let; 
Set X represent the set where the siblings x₁ and x₂ sit together
Set Y represent the set where the siblings y₁ and y₂ sit together
Set Z represent the set where the siblings z₁ and z₂ sit together
We have;
Where the three siblings don't sit together given as  ∩
∩ ∩
∩
By set theory, we have;
 =
 =  =
 =  
  =
 = 
Therefore;

Where;
 = The number of ways the 3 pairs of siblings can sit on the 7 chairs = 7!
 = The number of ways the 3 pairs of siblings can sit on the 7 chairs = 7!
 = The number of ways x₁ and x₂ can sit together on the 7 chairs = 2 × 6!
 = The number of ways x₁ and x₂ can sit together on the 7 chairs = 2 × 6!
 = The number of ways y₁ and y₂ can sit together on the 7 chairs = 2 × 6!
 = The number of ways y₁ and y₂ can sit together on the 7 chairs = 2 × 6!
 = The number of ways z₁ and z₂ can sit together on the 7 chairs = 2 × 6!
 = The number of ways z₁ and z₂ can sit together on the 7 chairs = 2 × 6!
 = The number of ways x₁ and x₂ and y₁ and y₂ can sit together on the 7 chairs = 2 × 2 × 5!
 = The number of ways x₁ and x₂ and y₁ and y₂ can sit together on the 7 chairs = 2 × 2 × 5!
 = The number of ways x₁ and x₂ and z₁ and z₂ can sit together on the 7 chairs = 2 × 2 × 5!
 = The number of ways x₁ and x₂ and z₁ and z₂ can sit together on the 7 chairs = 2 × 2 × 5!
 = The number of ways y₁ and y₂ and z₁ and z₂ can sit together on the 7 chairs = 2 × 2 × 5!
 = The number of ways y₁ and y₂ and z₁ and z₂ can sit together on the 7 chairs = 2 × 2 × 5!
 = The number of ways x₁ and x₂,  y₁ and y₂ and z₁ and z₂ can sit together on the 7 chairs = 2 × 2 × 2 × 4!
 = The number of ways x₁ and x₂,  y₁ and y₂ and z₁ and z₂ can sit together on the 7 chairs = 2 × 2 × 2 × 4!
Therefore, we get;
 = 7! - (2×6! + 2×6! + 2×6! - 2 × 2 × 5! - 2 × 2 × 5! - 2 × 2 × 5! + 2 × 2 × 2 × 4!)
 = 7! - (2×6! + 2×6! + 2×6! - 2 × 2 × 5! - 2 × 2 × 5! - 2 × 2 × 5! + 2 × 2 × 2 × 4!) 
 = 5,040 - 3072 = 1,968
 = 5,040 - 3072 = 1,968
The number of ways where the three siblings don't sit together given as  = 1,968
  = 1,968