In a certain country, private enterprises provide various goids and services. but the government provides free public transportation and electricity to its citizens. which type of economy exists in this country

Your question is a little ambiguous. Essentially you are describing what is known as a "mixed economy" where elements of a private enterprise economy are mixed with a centrally planned economy. A mixed economy like my own one (Ireland) has private industry which supplies the market with the goods it wants, while the government own the state transport company (CIE) which serve bus and rail services in Ireland. The state also has its own electricity company called ESB, however, both the transport industry and the electricity industry are open to competitors.

Hopefully I answered what I think you were asking, but if your question was essentially asking what type of economy is the opposite to private enterprise, your answer is a "Centrally Planned Economy" colloquially known as Communism. 

Hope I helped and please rate my answer :) 

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Cextra