First of all, $\text{TRIANGLE}$ has $8$ distinct letters, $3$ of which are vowels($\text{I, A, E}$) and rest are consonants($\text{T, R, N, G, L}$).
While attempting this, I came up with the idea of putting alternate vowels and consonants not to group same types together.
So, I decided to form two 'batteries'. [$\text{V}$ stands for Vowels and $\text{C}$ stands for consonants.]
$$\text{V} \text{ C}\text{ V} \text{ C}\text{ V} $$
And,
$$\text{C} \text{ V}\text{ C} \text{ V}\text{ C}$$
If we count all the permutations and then add them up (Mutually Exclusive Events), we can get total number of permutations.
Now, For the first case,
$3$ vowels can be arranged in the $3$ spaces required in $3! = 6$ ways
From $5$ consonants, $2$ spaces can be filled with consonants in $^5P_2 = 20$ ways
One battery, $(8 - 3- 2) = 3$ letters to arrange.
Total number of permutations : $6 * 20 * 4! = 2880$.
In Second case,
From $3$ vowels, $2$ spaces can be filled with vowels in $^3P_2 = 6$ ways
From $5$ consonants, $3$ spaces can be filled with consonants in $^5P_3 = 60$ ways.
One battery, $(8 - 2- 3) = 3$ letters to arrange.
Total number of permutations : $6 * 60 * 4! = 8640$
So, Total number of permutations for the word $\text{TRIANGLE} = 2880 + 8640 = 11520$
Again, My answer is incorrect, according to my textbook. They report $14400$ is the correct answer.
So, what did I miss here now? Please elaborate, and I'll be happy with any sort of help. [Seriously, this morning is getting even more hectic for me]