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## Homework Statement

Positive integers

*i, j, k, m,*and

*n*are randomly chosen (repetition is allowed) so that

2 <=

*i,j,k,m,n*<= 2009 . What is the probability that

*ijk + mn*is even?

## Homework Equations

## The Attempt at a Solution

Each of the positive integers is one out of 2008 numbers. 1004 of these numbers are even and 1004 of these numbers are odd. So, 50% chance of being an even number and 50% chance of being an odd number for each of the integers

*i,j,k,m,n*.

I looked at the

*ijk*term first and wrote out all possible combinations.

odd x odd x odd = odd

odd x odd x even = even

odd x even x odd = even

odd x even x even = even

even x odd x odd = even

even x odd x even = even

even x even x odd = even

even x even x even = even

1/8 possibilities are odd (12.5%) and 7/8 are even (87.5%)

I looked at the

*mn*term next and did the same.

odd x odd = odd

odd x even = even

even x even = even

even x odd = even

1/4 possibilities are odd (25%) and 3/4 are even (75%)

Next, I looked at the term

*ijk*and the term

*mn*added together.

odd + odd = even

odd + even = odd

even + odd = odd

even + even = even

2/4 possibilities are odd (50%) and 2/4 are even (50%)

I'm really not sure where to go from here. Am I on the right track at least? I'm just not sure how to combine the above facts into a statement about the probability of [

*ijk + mn*] being an even number. Any help is much appreciated