Considering that the Falklands war was a logistical nightmare for the British, and it was an attack upon British forces and an invasion on her territory, and aside from initial friction from the US, wouldn’t it be more convenient to do so?
The Parties agree that an armed attack against one or more of them in Europe or North America shall be considered an attack against them all …
Simply put, article 5 doesn’t apply since the Falklands is in the South Atlantic/South America.
Further, invoking article 5 would be an affirmative declaration of war that in theory would complicate matters severely from a diplomatic and governmental point of view.
Ultimately, the British decided to rely on Article 51 of the UN charter as their justification for the war due to it being an implied and automatic invocation vs an affirmative one.
This older answer by /u/k1990 explains it a lot better than I can: https://www.reddit.com/r/AskHistorians/comments/6e48mr/why_didnt_the_uk_declare_war_on_argentina_in/