Given a list of 8 numbers, there are exactly 8!different permutations of those 8

Question

Given a list of 8 numbers, there are exactly 8!different permutations of those 8 pieces. But, how many different permutations arethere between the numbers in which all even numbers are ineven slots and all odd numbers are in oddslots. {odd, even, odd, even, odd, even, odd,even}

For example, {1, 4, 7, 6, 3, 2, 5, 8} is a validexample. Given a list of 8 numbers, there are exactly 8!different permutations of those 8 pieces. But, how many different permutations arethere between the numbers in which all even numbers are ineven slots and all odd numbers are in oddslots. {odd, even, odd, even, odd, even, odd,even}

For example, {1, 4, 7, 6, 3, 2, 5, 8} is a validexample.

Explanation / Answer

The even slots are as follows A C E G you need to distribute the even numbers in the even slotswithout regard to the odd numbers obviously there are 4 x 3 x 2 x 1 ways to put the numbers2,4,6,8 into four slots
this endeavor has no bearing on the positioning of the oddslots, which are B D F H you need to distribute the odd numbers in these four slotswithout regard to the even numbers obviously there are 4 x 3 x 2 x 1 ways to put the odd numbers1,3,5,7 into four slots
for each way to put the even numbers, you can put the oddnumbers in any way you like. 4! x 4!
You might be interested to know that this is a common problem,and the general formula is 2ab/(a+b) when b elements of one kindand a elements of another are in a row and you need to know thenumber of unlike adjacent elements. If a=4 and b=4 then if randomly distributed the averageexpected number of adjacent elements is 32/8=4
you need to distribute the odd numbers in these four slotswithout regard to the even numbers obviously there are 4 x 3 x 2 x 1 ways to put the odd numbers1,3,5,7 into four slots
for each way to put the even numbers, you can put the oddnumbers in any way you like. 4! x 4!
You might be interested to know that this is a common problem,and the general formula is 2ab/(a+b) when b elements of one kindand a elements of another are in a row and you need to know thenumber of unlike adjacent elements. If a=4 and b=4 then if randomly distributed the averageexpected number of adjacent elements is 32/8=4

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