Question

4 cats and 4 dogs randomly standing in a line. Find out the no of patterns possible such that all dogs are not standing together?
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Answer 1

Answer:

Had there been fewer dogs, there would have been more possibilities. But, the number of dogs is exactly one more than that of the cats who’ll not allow them to be placed adjacently. The only arrangement pattern is this :

D-C-D-C-D-C-D-C-D

That is, the dogs must occupy odd positions and the cats-even. Now, once this is fixed, we can arrange the dogs amongst themselves in their even places (which is  4! ) and similarly the cats amongst their even places ( 4! ).

So, in totality, by product rule we have , total ways  =4!×4!  

=576

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