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.

5,760

Step-by-step explanation:

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. Let D represent the dogs and C represent the Cat , then we have

D C D C D C D C D D   or

D  D C D C D C D C D  or

D C DD C D C D C D or

D C D C D D C D C D or

D C D C D C DD C D

Each of the arrangements above  could be done in

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 )

= 2 x 24 x 24

= 1 , 152

The total arrangement = 5 x 1 , 152

= 5,7 60 ways

6mm^3 or .6cm^3

      216

      /   \

  36     6

  /   \     /

6     6   6

factor trees are nice.