The code word is TIE.
If you were told any one character of MOD, then you would not be able to
determine whether the number of vowels are one or two. e.g. if you were told M,
there are two words with M - AIM with 2 vowels and MOD with 1 vowel. So you
would not be able to say the number of vowels. Same arguments can be given
for characters O and D.
Hence, the word with any one of M, O or D is not a code word i.e. AIM, DUE,
OAT and MOD are not the code word. Thus, TIE is the code word.
T : two words - TIE and OAT, both with 2 vowels
I : two words - TIE and AIM, both with 2 vowels
E : two words - TIE and DUE, both with 2 vowels.