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Most phrases have very small percentages, as they are measured against all other combinations of words in literature.
Since it’s an informal idiom, it’s also much more common in spoken than in written language.
Even with those small percentages, ‘free as a bird’ was twice as common.
Exactly, only twice as common. To put in other words: For every two times someone says “free as a bird”, one person says “happy as a clam”.
That is much narrower than the gap between something commonly said and something rarely said.
Again, “happy as a clam” is very informal. “Free as a bird” is much less so, so it makes sense it pops up more often in literature.
Also more poetic, shows up in poetry and song lyrics, pre-written spreches, etc.
It’s also been used much longer. First known use of “happy as as a clam” was 1833 versus “free as a bird” being used in the 17th century.