Lonely Consonants

Let's take this in $2$ different cases

$\boxed{\text{Case 1 : When the word starts with a vowel}}\\ \text{The first letter can be filled in 3 ways, since there are 3 different vowels.} \\ \text{Since consonants must not occur together, in this word given, only way is to arrange vowel and consonants alternatively. Must also note that letters can repeat.} \\ \text{So second letter can be filled in 2 ways, as there are 2 consonants} \\ \text{3rd letter can be filled in 3 ways, as 3 there are vowels } \\ \text{4th letter can be filled in 2 ways, 2 consonants} \\ \text{5th letter can be filled in 1 way} \\ \therefore \text{Total no of different letters formed = } 3 \times 3 \times 2 \times 2 = 36\\ \boxed{\text{Case 2 : When the word starts with a consonant}}\\ \text{The first letter can be filled in 2 ways, since there are 2 different consonants.} \\ \text{So second letter can be filled in 3 ways, as there are 3 vowels} \\ \text{3rd letter can be filled in 2 ways, as 2 there are consonants } \\ \text{4th letter can be filled in 3 ways, 3 vowels} \\ \text{5th letter can be filled in 1 way} \\ \therefore \text{Total no of different letters formed = } 3 \times 3 \times 2 \times 2 = 36\\ \boxed{\therefore \text{Total no of ways to form different words = } 36 + 36 = 72}$

There are 2 consonants in the word (L, N), so there are two possible arrangements for them to be next to each other (LN, NL). For each of these, we have 4 different cases (either the 2-letter group is in positions 1-2, 2-3, 3-4 or 4-5), and for each of these 4 cases the remaining vowels can be rearranged in $3!=6$ ways.

So, the result is obtained by subtracting from all the possible rearrangements of the letters (5!) the ones that we listed above: $5! - 2*4*6 = 120 - 48 =\boxed{72}$