How many words can be formed using all the letters of the word ‘NATION’ so that all the three vowels should never come together?

Concept:

• Suppose a set of n objects has n₁ of one kind of object, n₂ of a second kind, n₃ of a third kind, and so on with n = n₁ + n₂ + n₃  +...+ n_k then the number of distinguishable  permutations of the n objects is  = \(\frac{{{\rm{n}}!}}{{{{\rm{n}}_1}! \times {{\rm{n}}_2}! \times {{\rm{n}}_3}! \ldots \ldots \ldots {{\rm{n}}_{\rm{k}}}!}}\)

Calculation:

The given word is 'NATION'

Number of letters repeating in the word =2N

Vowels in the word are {A, I, O}

Total number of words that can be formed from given word 'NATION' = \(\frac{{6!}}{{2!}} = 360\)

Consider A, I, O as one group

Then the no. of words formed by this group and remaining letters is 4!

The three vowels can be arranged among themselves in 3! Ways

∴ Number of words where all the vowels come together = \(\rm \frac {(4!) (3!)}{2}\) =72

Number of words formed so that all the three vowels are never together = Total number of words formed using all the letters in the word NATION − Number of words where all the vowels come together

Thus, the number of words formed so that all the three vowels are never together = 360 – 72 = 288