Given that there
are 17 portable mini suites (a.k.a. cages) in a row at the Paws and
Claws Holiday Pet Resort. They are neatly labeled with their guests'
names. There are 8 poodles and 9 tabbies. How many ways can the "suites"
be arranged if:
a) there are no restrictions.
b) cats and dogs must alternate.
c) dogs must be next to each other.
d) dogs must be next to each other and cats must be next to each other.
Part A:
If there are no restrictions, then the number of ways the "suites" can be arranged is given by:
[tex]17!=17\times16\times15\times14\times13\times12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1=355,687,428,096,000 \ ways[/tex]
Part B:
If cats and dogs must alternate, then the number of ways the "suites" can be arranged is given by:
[tex]9!\times8!=9\times8\times7\times6\times5\times4\times3\times2\times1\times8\times7\times6\times5\times4\times3\times2\times1=14,631,321,600 \ ways[/tex]
Part C:
If dogs must be next to each other, then the group of dogs are taken as 1 object and there are now 10 objects to arrange. The number of ways to arrange 10 objects is given by:
[tex]10!=10\times9\times8\times7\times6\times5\times4\times3\times2\times1=3,628,800
\ ways[/tex]
Also, the group of dogs can be arranged in:
[tex]8!=8\times7\times6\times5\times4\times3\times2\times1=40,320
\ ways[/tex]
Therefore, the total number of ways the "suites" can be arranged is given by:
[tex]10!\times8!=3,628,800\times40,320=146,313,216,000 \ ways[/tex]
Part D:
If dogs must be next to each other and cats must be next to each other, then the group of dogs are taken as 1 object and the group of cats are taken as 1 object, then there are now 2 objects to arrange.
The number of ways to arrange two objects is given by:
[tex]2!=2\times1=2[/tex]
The number of ways the 9 dogs are to be arranged is given by:
[tex]9!=9\times8\times7\times6\times5\times4\times3\times2\times1=362,880[/tex]
The number of ways the 8 cats are to be arranged is given by:
[tex]8!=8\times7\times6\times5\times4\times3\times2\times1=40,320[/tex]
Therefore, the total number of ways the "suites" can be arranged is given by:
[tex]2!\times9!\times8!=2\times362,880\times40,320=29,262,643,200 \ ways[/tex]
