Question

The number of words that can be formed with the letters of the word 'DEFINITE' if two vowels are together and the other two are also together but separated from the first two is

• A. 720
• B. 1680
• C. 1440
• D. 2880

Hint:

For arrangement questions with groups, arrange consonants first and use gaps to place vowel groupsThis helps maintain separation conditions easily.

Answer 1

The Correct Option is C

Step 1: Identify vowels and consonants.
The word is:
\[ DEFINITE \]
Vowels are:
\[ E,I,I,E \]
Consonants are:
\[ D,F,N,T \]
There are 4 vowels and 4 consonants.

Step 2: Arrange the consonants first.
The 4 consonants \(D,F,N,T\) can be arranged in:
\[ 4! = 24 \]
ways.

Step 3: Find gaps around consonants.
After arranging 4 consonants, there are 5 gaps:
\[ \_ C \_ C \_ C \_ C \_ \]
The two vowel-pairs must be placed in two different gaps so that they remain separated.

Step 4: Select gaps for two vowel-pairs.
Number of ways to choose 2 gaps from 5 gaps:
\[ \binom{5}{2} = 10 \]

Step 5: Divide vowels into two pairs.
The vowels are \(E,E,I,I\).
The possible separated pairs are:
\[ EE \text{ and } II \]
or
\[ EI \text{ and } EI \]
Considering arrangements within pairs and placement of pairs gives total vowel-pair arrangements as:
\[ 6 \]

Step 6: Calculate total number of words.
\[ \text{Total words} = 4! \times \binom{5}{2} \times 6 \]
\[ = 24 \times 10 \times 6 \]
\[ = 1440 \]

Step 7: Final conclusion.
\[ \boxed{1440} \]