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I would be very grateful if you could give some rules to know when I have to use, North England or the North of England. I need to know when the article is needed and when it is not.

Thank you for your help.
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not a hard and fast rule, but ... if there is an administrative region/city/etc. whose official name includes a direction, you should use it with the place name and capitalize it as well (North America, New South Wales, etc.). if you are speaking generally of a fairly non-specific area, do not capitalize (the north of France). for this second group, you may especially consider using the adjectives northern, southern, eastern, and western, though these words also may be "official" designations of countries, town, etc. (Northern Ireland).

as for western Europe ... or Western Europe ... that is a tricky one. i would say the first responds to the western part/regions of Europe, where the second more or less corresponds to a somewhat (sorry so vague here) list of countries. context comes into play here, as well as the general understanding of your readers (e.g. is Norway part of Western Europe even though it is generally considered a northern country?)