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A partnership is formed when two or more individuals (which can include legal entities, such as limited companies) agree that they form a partnership. There are no other formalities that need to occur.
However, without a 'Partnership agreement' stating the rules and responsibilities within a partnership, all the rules of the partnership will be set out by the 1890 Partnership Act. As the majority of partnerships prefer to operate under their own rules, they use a partnership agreement to formalise the structure in the way they want to run their business.