How to find the regular expression?

I need a regular expression which will describe the strings of the alphabet {a, b, c} of the language {a^n b^m c^k, where n + m + k is even for variable integers n, m, k ≥ 0}. For example, the strings ε, aabc, abbccccc belong to the language, while the strings abccaa, a, aaabbcc
do not belong.

Should I build first a Finite Automaton or is it possible without it?

Any help will be appreciated!

>Solution :

Interpreting this as a programming question (which is what Stack Overflow is about), you could use this regular expression:

^(?=(..)*$)a*b*c*$

Explanation:

• ^ and $ match the start/end of the input.
• (..)* matches a series of pairs of any characters, so it always matches an even number of characters.
• (?=(..)*$) is a look-ahead expression that verifies that the total length is even.
• a*b*c* matches any number of a, followed by any number of b, followed by any number of c.

Without look-ahead, you’d need to foresee the cases where all groups are even, or just one group is even (and the two others odd):

^(aa)*((bb)*(bc)?(cc)*|a(bb)*[bc](cc)*)$