Trodd Numbers

If a sum of three numbers is odd then either these three numbers are all odd or one of them is odd and the other two are even. As such, $n$ will have one of these forms: $ooooooo$ , $eeoeeoe$ , $oeeoeeo$ or $eoeeoee$ , with $e$ being an even digit and $o$ an odd one. Since there are only three odd numbers between $0$ and $6$ and only $4$ even ones, we can conclude $n$ is of the form: $oeeoeeo$ . So, since there are $4!$ permutations between the four even digits and $3!$ between the odd ones, there are $4! * 3! = 144$ numbers which fullfill the condition given. Another way to calculate the possible combinations is to start with the first digit and recognizing that there are $3$ possible digits for it, $4$ for the next one, $3$ for the next one ( given that one of the even digits has already been used) and so on, which yields the same answer: $3 * 4 * 3 * 2 * 2 * 1 * 1 = 144$ .