# There is exactly one possible order in which the cities are used if which one of the following is true?

Luke on February 14, 2020

Options D & E

I'm not understanding the difference between options D&E, it seems like either forces you into the same one sequence, how do you differentiate them?

shunhe on February 14, 2020

Hi @Luke,

Thanks for the question! First, letâ€™s take a look at answer (D). (D) tells us that V is used in the first slot.

V _ _ _ _ _

Well, based on rule 2, we know W must be in the second slot, and based on rule 4, we know that L must be in the fourth slow.

V W _ L _ _

But now, based on rule 1, we know that T is in the third slot.

V W T L _ _

Finally, based on rule 3, we know that M is in the last slot, leaving N for the second-to-last slot.

V W T L N M

And so we know that (D) is right because it leaves open only this order. Now letâ€™s take a look at (E). Letâ€™s put W in the second slot.

_ W _ _ _ _

Now, notice that one possibility is the slot we just derived, so we know thereâ€™s at least that possibility. But based on rule 2, V can go before or after W. Letâ€™s try putting V after W and see if we can make any boards that are consistent with the rules.

_ W V _ _ _

Well, we know that V and L are separated by two, so L goes in the last slot.

_ W V _ _ L

We also know that M and T differ by 2, which means one of them has to be first, and the other fourth. Then we can put N in fifth. So we can actually make another two game boards at least:

M W V T N L

T W V M N L

And since these are consistent with the rules, thereâ€™s more than one possible order with (E), and so (E) is wrong. Hope this helps! Feel free to ask any other questions that you might have.