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Q.
The number of ways in which the letters of word MEDICAL can be arranged such that $A$ and $E$ are together but all the vowels never come together is ____.
Permutations and Combinations
Solution:
We have letters $M , D , C , L , E , I , A$
We have $4$ consonants, which can be arranged in $4 !$ ways.
Now, there are five places, where we have to arrange vowels $A , E , I$ so that $A$ and $E$ are together but all are not together.
So, we have to arrange $AE$ and $I$.
Number of ways are ${ }^{5} C_{2} \times 2 ! \times 2 !$ (as AE can also be arranged in $2 !$ ways)
So, total number of arrangements $=4 ! \times{ }^{5} C_{2} \times 2 ! \times 2 !$
$= 960 $