Question

Number of different words that can be formed using all the letters of the word $DEEPMALA$ if two vowels are together and the other two are also together but separated from the first two is

• A. $960$
• B. $1200$
• C. $2160$
• D. $1440$

Answer 1

The correct option is D $1440$

Number of possible arrangements of vowels $=\frac{4!}{2!2!}=6$

Now, we have to make the cases of how this can be arranged.

Case$1:$ In first place, there is a $2$ vowel.
$\underset{–}{V}\underset{–}{}\underset{–}{}\underset{–}{}\underset{–}{}\underset{–}{}$ This can be formed in $\mathrm{1.4.4.3.2.1}$ ways.

$\Rightarrow$ Number of ways$=1\Rightarrow 1\times 4\times 4\times 3\times 2\times 1=96$

Case$2:$ In second place there is a $2$ vowel
$\underset{–}{V}\underset{–}{}\underset{–}{}\underset{–}{}\underset{–}{}\underset{–}{}$ This can be formed in $\mathrm{4.1.3.3.2.1}$ ways.

$\Rightarrow$ Number of ways $=4\times 1\times 3\times 3\times 2\times 1=72$

Case$3:$ In third place there is a $2$ vowel.

$\underset{–}{V}\underset{–}{}\underset{–}{}\underset{–}{}\underset{–}{}\underset{–}{}$ This can be formed in $\mathrm{4.3.1.2.2.1}$ ways.

$\Rightarrow$ Number of ways $=4\times 3\times 1\times 2\times 2\times 1=48$

Case$4:$ In fourth place there is a $2$ vowel.
$\underset{–}{V}\underset{–}{}\underset{–}{}\underset{–}{}\underset{–}{}\underset{–}{}$ This can be formed in $\mathrm{4.3.2.1.1.1}$ ways.

$\Rightarrow$ Number of ways $=4\times 3\times 2\times 1\times 1\times 1=24$

No more possible case will be there as the letter will be repeated.
Thus, total ways $=96+72+48+24=240$

Now, vowel can be arranged in $6$ ways.
Therefore, number of different words $=240\times 6=1440$