How many words, with or without meaning, can be formed using all the letters of the word $\mathrm{EQUATION}$ at a time so that the vowels and consonants occur together?
How many words, with or without meaning, can be formed using all the letters of the word $\mathrm{EQUATION}$ at a time so that the vowels and consonants occur together?
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 $^{2} P_{2}=2 !$
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 $2! \times 5! \times 3!=1440$
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