PERMUTATIONS HELP: How can you arrange 9 people if Tina has to be next to Quinton OR Tatiana?

Question

The answer is 151,200 but I am not getting that

Answer 1

The problem is an application of the inclusion-exclusion principle.

The permutations where Tina and Quinton are next to each other are 2*8!. Indeed, the 8! is form considering Tina and Quinton as a unit, and the 2 is from the two possible orders: Tina then Quinton versus Quinton then Tina.

Similarly, the permutations where Tina and Tatiana are next to each other are also 2*8!.

Now we need to exclude the permutations where Tina is next to both Quinton and Tatiana. Those are 2*7! since we consider all three to be a unit, and we can arrange the triple as Quinton, Tina, and Tatiana, or as Tatiana, Tina, and Quinton.

So, by the inclusion-exclusion principle, the final answer is 2*8! + 2*8! - 2*7! = 151200.

Answer 2

Consider the two as a unit

Tina and Quinton (in that order) x 7!

Quintin and Tina (in that order) x 7!

Tina and Tatiana (in that order) x 7!

Tatiana and Tina (in that order) x 7!

Is the sum of those 151,200?