How many words, with or without meaning, can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together?

Explanation

In the word EQUATION, there are 5 vowels, namely, A, E, I, `O`, and U, and 3 consonants, namely, `Q, T`, and `N`.


Since all the vowels and consonants have to occur together, both (AEIOU) and (QTN) can be assumed as single objects. Then, the permutations of these 2 objects taken all at a time are counted. This number would be `2_(P_2)=>2` !


Hence, by multiplication principle, required number of words
`=>2 ! times 5` ! `times 3 !`


`=>1440`