We know that all vowels are in the front and consonants are at the end, so I broke it down like 2 different words.
I would have thought we have to add the 2 permutations together (because the vowels and consonants will not change places with each other as groups), which would give: (3!5!)/2! = 360
I can see that you want to just understand why we are performing a multiplication instead of an addition.
When the question said total ways the letters can be arranged, it means that how many different words can be formed.
To better understand the reason for multiplication, let us take a simpler case where we have these letters EOOTXX. Here EOO has 3 ways, and TXX has 3 ways as well.
Ignoring the thought of whether to add or multiply, just for the time being to understand the approach, let us just first consider EOO as is. With EOO, all the permutations of TXX can form new word, i.e.:
EOO + TXX = EOOTXX
EOO + TXT = EOOTXT
EOO + XTT = EOOXTT
Similarly 3 each with OEO and OOE
So as you can see that for each EOO permutation, we have 3 ways we can put “TXX”. And we have 3 such permutations for EOO. So: EOO with 3 TXX + OEO with 3 TXX + OOE with 3 TXX. Which can be interpreted as (3 ways of EOO) x (3 ways of TXX). And that is MULTIPLICATION.
