Find the number of different words that can be formed from the letters of the word ‘TRIANGLE’ so that no vowels are together

Question

I have a question that says: Find the number of different words that can be formed from the letters of the word ‘TRIANGLE’ so that no vowels are together

I did this in the following way: Number of ways in which TRIANGLE can be arranged - Number of ways in which TRIANGLE can be arranged where 3 vowels are together - Number of ways in which TRIANGLE can be arranged where 2 vowels are together = 8! - (6!*3!) - (7!*2!*3) = 5760

The correct answer in my book is given to be 14400 calculated as: xTxRxNxGxLx : x depicts spaces where vowels can be arranged and they are not together. Therefore, Number of ways in which Consonants can be arranged*Number of ways in which vowels can be arranged = 5!6C33! = 14400.

Can anyone help me figure out why the method I follow isn't working?

Answer 1

Applying inclusive - exclusive principle

Making group with two vowels$(7!\times 2!\times 3)$ will cover also making group with three vowels $(6!\times3!)$

for example $TR(IA)ENGL$ arrangements of $(IA)$ together also has $E(IA)$ and $(IA)E$

$$8! - (7!\times 2!\times 3) + (6!\times3!)= 14400$$