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It may be organic, may be inorganic but is ionic and binary in nature and contains carbon. Is there any such compound?

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The well known compound calcium carbide is an ionic compound containing carbon (as $\ce{C2 ^2-}$ units).

The compound was once very common as its reaction with water produces acetylene which was used in lamps and even early car headlights. It is also used in some chemical processes.

The invention of an easy way to make it in the late 19th century was a major part of the industrial revolution in the chemical industry.

It is probably the best known ionic compound containing carbon.

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Aluminum carbide, $\ce{Al4C3}$, is ionic and reacts strongly with water according to : $\ce{Al4C3 + 12 H2O -> 4 Al(OH)3 + 3 CH4}$

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