The phrase "give a damn" is an idiomatic way of saying that someone is concerned or has an interest in something, or that they care about it.
It's most often used in the negative, where it emphasizes a lack of concern or indifference. For example, "I don't give a damn what you think" means "I don't care what you think at all".
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The expression "give a damn" has a relatively long history. It may have originated from the phrase "I don't give a damn," which was used in America in the 1890s, according to fofweb.infobase.com. The word "damn" in this context can refer to a small amount or something insignificant, according to English Language & Usage Stack Exchange.
Alternatives:You can often use similar expressions like "don't care a hoot," "don't care a fig," or "don't care a hang" to convey the same meaning.
Informal and Rude:"Give a damn" is considered an informal and somewhat rude expression,