How many 5-letter words can be formed from the letters of the word mechanical that word always starts with a consonant?

How many permutations can be formed by the letters of the word, 'VOWELS', wheni there is no restriction on letters?ii each word begins with E ?iii each word begins with 0 and ends with L ?iv all vowels come together?v all consonants come together?

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The miscalculation is here:

the second case was where there were two groups of two alike letters and one different letter. Thus, the number of words formed were: 2C1 . 4C2 . 3! = 72

I assume your ${2\choose 1}$ implies you are choosing from $G$ and $T$ and ${4\choose 2}$ implies you are choosing 2 letters from $M,N,A,E$. Your ${4\choose 2}$ is correct, since you will get two groups of two alike letters, consequently total $4$ letters. However, your ${2\choose 1}$ is not correct (it must be ${4\choose 1}$), because you can choose not only from $G$ and $T$, but also from the remaining two letters of $M,N,A,E$. Because, a single letter will be a different (unlike) letter.

All the rest is fine.