Hi kluevehe
As the problem says, a and b should be integer and a+b=5, let us first find set of integers which solve the equation a+b=5.
For ex it can be a=1,b=4 or a=4,b=1.. next a=2,b=3 or a=3,b=2
Lets take into account negative integers too. a=6,b=-1
Taking all the above set of integers which solve the equation a+b=5
Lets check the first option i.e The product of a and b is odd. Now if you take a=1,b=4 or a=4,b=1 The product is 4 which is even. So eliminate this option
Next option is If a is odd, b is even.. Now if we take this set of integers a=1,b=4 a is odd, b is even.
c option can be eliminated out, because the first option is ruled out.
d option, i,e II and III is the right option because even this condition i.e if a is negative, b is positive is true for negative integers ( a = -1 or b =6)
Hoping that you understand the above solution.