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That is correct! The piano (normally) is not included in the "trio" or "quartet" count.
a violin, viola and a cello or two violins and a cello
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Hans Otte has written:
'Trio for violin, viola, and 'cello' -- subject(s): String trios, Scores
A "trio" ... consists of string instruments plus a piano. The piano is a member of the percussion family.