There are 11 portable mini suites (a.k.a. cages) in a row at the Paws and Claws Holiday Pet Resort. They are neatly labeled with

Question

2 months ago

7

There are 11 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 5 poodles and 6 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.

Mathematics

1 answer:

AnnZ [12.3K] · Answer 1 · 2026-03-20T02:24:55+03:00

Step-by-step explanation:

a) 7!

In absence of any restrictions, the answer is 7! as it represents the permutations of all animals.

b) 4! x 3!

Considering there are 6 cats and 5 dogs, the first and last slots must be occupied by cats to ensure alternate arrangements. The only options available then are based on the arrangement of the cats among themselves and the dogs among themselves, yielding 4! permutations for the cats and 3! for the dogs, thus leading to a total of 4! x 3! arrangements.

c) 3! x 5!

Here, the arrangement of the dogs among themselves can occur in 3! ways. Considering the dogs as a singular “object,” we can arrange this unit with the 4 cats, providing 5! total arrangements possible, leading to 3! · 5! arrangement possibilities.

d) 2 x 4! x 3!

In this scenario, both cats and dogs must be grouped together, allowing positions where all cats come before the dogs or vice versa. As there are two configurations, the resultant count is 2 multiplied by both arrangements, resulting in 2 x 4! x 3!