How many different words can be formed by using all the letters $$o($$ the word 'ALLAHABAD' such that vowels occupy the even positions, is

Question

How many different words can be formed by using all the letters $$o($$ the word 'ALLAHABAD' such that vowels  occupy the even positions, is

Infinity Learn · Accepted Answer

There are $$4$$ vowels and all are alike i.e. $$4A$$'s. Also, there are $$4$$ even places viz $$2nd$$, $$4th$$, $$6th$$ and $$8th$$. So, these $$4$$ even places can be occupied by $$4$$ vowels in $$4$$!$$4$$!$$=1$$ way. Now, we are left with $$5$$ places in which $$5$$ letters, of which two are alike (2$$ L'$$s) and other distinct, can be arranged in $$5$$!$$2$$! ways.Hence, the total number of words in which vowels occupy the even places $$=5$$!$$2$$!×$$4$$!$$4$$!$$=5$$!$$2$$!$$=60$$

How many different words can be formed by using all the letters o( the word 'ALLAHABAD' such that vowels occupy the even positions, is

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