Question

How many words can be formed using all the letters of the word 'NATION' so that all the three vowels should never come together?
A) 354
B) 348
C) 288
D) None of the above

Answer 1

Permutations are the different ways in which a collection of items can be arranged. For example:
The different ways in which the alphabets A, B and C can be grouped together, taken all at a time, are ABC, ACB, BCA, CBA, CAB, BAC. Note that ABC and CBA are not the same as the order of arrangement is different. The same rule applies while solving any problem in Permutations.
The number of ways in which n things can be arranged, taken all at a time, ${}^n{P_n}{\text{ }} = {\text{ }}n!$, called ‘n factorial.’

Complete step-by-step answer:
Total number of letters in “NATION” = 6.
So, number of words possible with the combination of these letters =$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$
As, there is 2 N in the word, so we have to divide the number of words value by 2= $\dfrac{{720}}{{2!}} = 360$
Therefore, total number of words that can be formed using the letters in the word 'NATION'=360
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
Vowels in the word are A,I,O
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
As, there is 2 N in the word, so we have to divide the number of words value by 2
Hence, Number of words where all the vowels come together $= \dfrac{{\left( {4!} \right)\left( {3!} \right)}}{{2!}} = 72$
Thus, the number of words formed so that all the three vowels are never together $= 360 - 72 = 288$
Therefore, 288 words can be formed using all the letters of the word 'NATION' so that all the three vowels should never come together

So, option (C) is the correct answer.

Note: Always keep an eye on the keywords used in the question. The keywords can help you get the answer easily.
The keywords like-selection, choose, pick, and combination-indicates that it is a combination question.
Keywords like-arrangement, ordered, unique- indicates that it is a permutation question.
If keywords are not given, then visualize the scenario presented in the question and then think in terms of combination and arrangement.