Question

in how many different ways can the letter of the word trainer be arranged so that the vowels always come together

Answer 1

Given word : - TRAINER

vowels(4) = I , A , E

Consonant (4) = T,R,N,R

If we take vowels together , it can be treated as one letter.

Total letters - 1 (I,A,E) + 4(T,R,N,R) = 5

Number of ways to arrange 5 letters = $^{5}P_5 = 5! = 120 \ ^{*}(Note)$

Again,

vowels can also be arranged in 3! = 6 ways

According to fundamental concept of calculation, if first event happens in m ways and second event happens in n ways; So if both events happens together, then number of ways = mn

Similarly,

Number of ways = 120* 6 = 720 ways

* Note: In some cases 120 can also be considered as a answer because it is not mention that whether we have to change place of vowels of not.