$EQUATION$ शब्द के अक्षरों से कितने, अर्थपूर्ण या अर्थहीन, शब्दों की रचना की जा सकती है, जबकि स्वर तथा व्यंजक एक साथ रहते हैं ?

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 $2! \times 5! \times 3!=1440$