Question

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?

Answer 1

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 $^2P_2=2!$
Corresponding to each of these permutations, there are 5! permutations of the five vowels taken all at a time and 3! permutations of the 3 consonants taken all at a time. 
Hence, by multiplication principle, required number of words = $2! \times 5! \times 3! = 1440$