The state or quality of being extremely poor and lacking basic necessities.
From 'destitute' plus the suffix '-ness,' which converts adjectives into abstract nouns representing qualities or states. This is one of English's most common word-formation patterns.
The '-ness' suffix is so productive that you can almost add it to any adjective ('blueness,' 'kindness,' 'destituteness'), yet we rarely use the abstract noun form when we could say 'destitution'—a reminder that grammar allows many paths, but speakers choose the well-worn ones.
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