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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
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B
1200
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C
2160
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D
1440
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Solution

The correct option is D $$1440$$
Number of possible arrangements of vowels $$=\displaystyle\frac { 4! }{ 2!2! } =6$$

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

Case$$1:$$ In first place, there is a $$2$$ vowel.
$$\underline { V } \underline {  } \underline {  } \underline {  } \underline {  } \underline {  } $$ This can be formed in $$1.4.4.3.2.1$$ 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
$$\underline { V } \underline {  } \underline {  } \underline {  } \underline {  } \underline {  } $$ This can be formed in $$4.1.3.3.2.1$$ 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.

$$\underline { V } \underline {  } \underline {  } \underline {  } \underline {  } \underline {  } $$ This can be formed in $$4.3.1.2.2.1$$ 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.
$$\underline { V } \underline {  } \underline {  } \underline {  } \underline {  } \underline {  } $$ This can be formed in $$4.3.2.1.1.1$$ 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 $$=240\times 6=1440$$ 

Maths

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