B seems to be right.
Use of between is strictly restricted to compare two things. Between X and Y is correct idiomatic form. Choice B uses the correct form of comparison. Also notice ayreâ€™s lecture requires sigular form "illustrates" which is present in choice B.
Dr. Sayreâ€™s lecture recounted several little-know episodes in the relations between nations that illustrates what is wrong with alliances and treaties that do not have popular support.
(A) relations between nations that illustrates
(B) relation of one nation with another that illustrates
(C) relations between nations that illustrate
(D) relation of one nation with another and illustrate
(E) relations of nations that illustrates
Is it an idiom relation of X with Y or relation between X and Y