How many arrangements of the numbers 1,2,3,4,5,6,7,8,9 are there where the sum of any two adjacent numbers is odd?
+23252 For two adjacent numbers to have a odd sum, one of the numbers must be odd and the other must be even.
So, the numbers must be arranged as: O - E - O - E - O - E - O - E - O
There are 5 choices for the first odd number.
There are 4 choices for the first even number.
There are 4 choices for the second odd number.
There are 3 choices for the second even number.
There are 3 choices for the third odd number.
There are 2 choices for the third even number.
There are 2 choices for the fourth odd number.
There is 1 choice for the fourth (and last) even number.
There is 1 choice for the fifth (and last) odd number.
Multiplying these choices together: 5 x 4 x 4 x 3 x 3 x 2 x 2 x 1 x 1 choices.
