Permutations Integers 1 to 9 all even numbers stay in their natural positions I have these questions regarding Inclusion-Exclusion Principle two of them have been bothering me. The questions goes:
How many ways can we write the numbers 1 to 9 [1,2,3,4,5,6,7,8,9] if:

*

*All even numbers stay in their natural positions

*No even numbers stay in their natural positions

*All odd numbers stay in their natural positions

*No odd numbers stay in their natural positions

I solved no. 2 and 4 using the principle of inclusion-exclusion but haven't been able to solve 1 and 4. Should the answers of 2 and 4 be of help?

To 1:
You have five places where you can put the five odd numbers (1, 3, 5, 7 and 9):
_ 2 _ 4 _ 6 _ 8 _
So you don't need to bother about the even numbers. The number of different permutations will be: 5! = 120
To 3:
You have four places where you can put the four even numbers (2, 4, 6 and 8):
1 _ 3 _ 5 _ 7 _ 9
So you don't need to bother about the odd numbers. The number of different permutations will be: 4! = 24
