If a, b, and c are integer and ab + c is odd, which of the m

They've listed all possible scenarios for a, b, and c. As each of them can be either even or odd (2 option for each) then there are total of 2*2*2=8 scenarios possible. Then they evaluated each, to see which scenario makes ab+c odd and after that picked these scenarios to work with options I, II, and III.

Personally I'd approach this question in another way.
\(ab+c=odd\), means:
1. \(ab=even\) (which means that at least one of them is even) and \(c=odd\);
OR:
2. \(ab=odd\) (which means that both are odd) and \(c=even\).

We can see that in both cases at least one is even thus option III (\(abc=even\)) is always true.

Now, for I: if \(c=odd\) (case 1) and \(a=odd\) (so \(b=even\)) then \(a+c=even\neq{odd}\), so this option is not always true.

Exactly the same for option II: if \(c=odd\) (case 1) and \(b=odd\) (so \(a=even\)) then \(b+c=even\neq{odd}\), so this option is not always true.

Answer: C (III only).

Hope it's clear.