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

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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