Questions & Answers

Question

Answers

$\left( a \right)$ 18720

$\left( b \right)$ 5040

$\left( c \right)$ 9360

$\left( d \right)$ 10080

Answer
Verified

Given word:

‘NATIONAL’

As we see that all the letters in the given word are different except 1 and the number of letters are 8.

As we all know there are 5 vowels present in the English alphabets which are given as A, E, I, O and U.

Out of these alphabets the number of alphabets are present in the given word NATIONAL are A, I and O and A, as A is repeated two times

So in the given word there are 4 vowels and 4 consonants.

Now the total number of words which are possible form the letters of a given word NATIONAL = $\dfrac{{8!}}{{2!}}$, as A is repeated one time so divide by 2!

Consider that all the vowels are together so consider four vowels present in the given word as one letter.

So the arrangements of vowels internally = $\dfrac{{4!}}{{2!}}$

So there are 5 letters in the word so the number of arrangements = 5!

So the total number of words possible when all the vowels are together = \[\left( {5! \times \dfrac{{4!}}{{2!}}} \right)\]

So the total number of words such that no vowel is together is the difference of total number of words from the given word and the total number of words when all the vowels are together.

Therefore, the total number of words such that no vowel is together = \[\dfrac{{8!}}{{2!}} - \left( {5! \times \dfrac{{4!}}{{2!}}} \right)\]

Now simplify we have,

\[ \Rightarrow \dfrac{{8.7.6.5!}}{{2.1}} - \left( {5! \times \dfrac{{4.3.2.1}}{{2.1}}} \right)\]

\[ \Rightarrow 4.7.6.5! - \left( {12.5!} \right)\]

\[ \Rightarrow 5!\left( {168 - 12} \right) = 120\left( {156} \right) = 18720\]