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Sagot :
Using the arrangements formula, the number of orders is given as follows:
- 39,916,800 if no restrictions.
- 86,400 if they are lined up alternatively.
- 7,257,600 if the first and last must be cats.
What is the arrangements formula?
The number of possible arrangements of n elements is given by the factorial of n, that is:
[tex]A_n = n![/tex]
When there are no restrictions, the number of ways is:
[tex]A_{11} = 11! = 39,916,800[/tex]
When they must be lined alternatively, the 6 dogs can be arranged in 6! ways, and the 5 cats in 5! ways, hence the number of orders is:
[tex]A_6A_5 = 6! \times 5! = 86,400[/tex]
When the first and last are cats, we have that:
- For the first and last animals, there are 5!/2! = 20 ways.
- For the middle 9 animals, there are 9! ways.
Hence:
20 x 9! = 7,257,600.
More can be learned about the arrangements formula at https://brainly.com/question/24648661
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