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.
Part A:
If there are no restrictions, then the number of ways the "suites" can be arranged is given by:
Part B:
If cats and dogs must alternate, then the number of ways the "suites" can be arranged is given by:
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:
Also, the group of dogs can be arranged in:
Therefore, the total number of ways the "suites" can be arranged is given by:
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:
The number of ways the 9 dogs are to be arranged is given by:
The number of ways the 8 cats are to be arranged is given by:
Therefore, the total number of ways the "suites" can be arranged is given by:
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:
Part B:
If cats and dogs must alternate, then the number of ways the "suites" can be arranged is given by:
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:
Also, the group of dogs can be arranged in:
Therefore, the total number of ways the "suites" can be arranged is given by:
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:
The number of ways the 9 dogs are to be arranged is given by:
The number of ways the 8 cats are to be arranged is given by:
Therefore, the total number of ways the "suites" can be arranged is given by: