Regular expression for $\{a^k b^m c^n \mid k+m+n \text{ is odd} \}$

I have to make a regular expression from the following laguage:


{$a^kb^mc^n : $ where k + m + n is odd}

Is is possible for the sum of three numbers to be odd (other than three consecutive odd numbers)?

I have this so far:


{(abbbccccc) + (abbbbbccc) + (aaabccccc) + (aaabbbbbc) + (aaaaabccc) + (aaaaabbbc)}

but I am realizing that there are way more possibilities to this pattern... How can I formulate a string that encompasses all of this?

What you need is that either exactly two of $k$, $m$, $n$ even, or all three odd, because two odds and an even make an even; and three evens make an even.

An odd number of $a$'s translates to the regular expression $a (a a)^$, an even number to $(a a)^$.

Pulling the above together:
$$
a (a a)^ (b b)^ (c c)^ \mid (a a)^ b (b b)^ (c c)^ \mid (a a )^ (b b)^ c (c c)^ \mid a (a a)^ b (b b)^ c (c c)^
$$

StackExchange Computer Science Q#9478, answer score: 7