Translation:He is going to live in Mexico for a year.
I have always been taught in Spanish classes that this should be Durante un año, or just un año, not por, and including por is a common mistake that non-native speakers make.
This is from a language site called "Stackexchange": I think that the simplest explanation it that "por" means that something has happened during all the extension of the time period mentioned, whereas "durante" points to some occasions during that period:
Durante febrero, las condiciones empeoraron: At some point in February, the conditions got worse, but not necessarily continuously... perhaps the rest of the month was OK but on the 20th they really got worse. Durante dos años trabajé allí ocasionalmente: I worked there as a temp, on and off.] Trabajé allí por dos años: every day, 9 to 5. Durante años los científicos pensaron esto: they thought that, but only on the occasions when they gave some attention to the matter. Por años los científicos pensaron esto: they thought it continuously, with no change of opinion.
That been said, there is still a restriction that "por" needs to be used with indeterminate periods (never mind plural/singular): por años por tres años por cinco horas pero nunca por el miércoles, por 1980 (could be used but in the sense of "around 1980"). Durante can be used in all of those cases: durante años, durante 1980.
Por versus para is very challenging as far as when you use them, as well as the fact that they both have so many meanings. I guess in time I will get used to it and just have to except it.
Your second choice is simply a way to demonstrate the pronounciation.
This 'select the boxed words' segment -' he is going to live in Mexico for a year'... was it just happenstance the remaining four blocks read as ' his wife soon happy' ? :-)