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How many ways are there for six dogs and four cats to sit in a row in a strawberry field so that no two cats sit next to each other? Each dog is distinct from other dogs and each cat is distinct from other cats.Hint: First position the dogs and then consider possible positions for the cats.

Respuesta :

bayosanuade

Answer: 5,760

Step-by-step explanation:

[tex]\\[/tex]It was given that each dog is distinct from other dogs and each cat is distinct from other cats, Also from the hint given, the First position is for the dogs. [tex]\\[/tex]Let D represent the dogs and C represent the Cat , then we have

[tex]\\[/tex]D C D C D C D C D D   or

[tex]\\[/tex]D  D C D C D C D C D  or

[tex]\\[/tex]D C DD C D C D C D or

[tex]\\[/tex]D C D C D D C D C D or

[tex]\\[/tex]D C D C D C DD C D

[tex]\\[/tex]Each of the arrangements above  could be done in

[tex]\\[/tex]2 x 4! x 4!  ( it is constant that D is starting , so I am only left with the arrangement of the remaining 5 D's  , out of the remaining 5 D's it is also constant that tow of them will be together and this could be done in 2 ways, so I have 4! left for the D's and 4! also for the C's )

[tex]\\[/tex]= 2 x 24 x 24

[tex]\\[/tex] = 1 , 152

[tex]\\[/tex]The total arrangement = 5 x 1 , 152

[tex]\\[/tex]= 5,7 60 ways

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