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The phrase "amalgamating the three" is correct and usable in written English.
It can be used when discussing the process of combining or merging three distinct elements into a single entity or concept.
Example: "The project involves amalgamating the three different proposals into one comprehensive plan."
Alternatives: "merging the three" or "combining the three".
Exact(2)
The first of its two parts, equivalent to a symphonic first movement, is a setting of the medieval Roman Catholic Pentecost hymn "Veni Creator Spiritus"; part two, amalgamating the three movement-types of the traditional symphony, has for its text the mystical closing scene of J.W. von Goethe's Faust drama (the scene of Faust's redemption).
The final unit assigned to the brigade was the Imperial Service Machine-Gun Squadron formed on 10 June 1918 by amalgamating the three cavalry regiment's machine-gun sections into one unit.
Similar(58)
It is easy to see the arguments for amalgamating the two.
Paschal Donohoe, the finance minister, will this morning circulate the terms of reference for a new working group that will set about amalgamating the two taxes.
It would accept that New Labour has absorbed most of what Lib Dems value, and move gradually towards amalgamating the two.
249 proposed for amalgamating the six battalions employed in the Myingyan, Pakokku, Minbu, Magwe, Lower Chindwin, and Sagaing districts into three battalions, thereby effecting a reduction of ten and a half companies, by which another regular regiment was to be constituted.
Indeed, experimental results illuminate and corroborate these theoretical findings, demonstrating that excellent results are achieved in both benchmark and real data problems by amalgamating the two approaches.
Although the results were qualitatively similar, there is a problem that prevents us from simply amalgamating the two data sets.
The results are presented by amalgamating the seven categories into three groups: affluent (categories 1 and 2), intermediate (categories 3 5) and deprived (categories 6 and 7).
This is done by physically amalgamating the two distinct embryos into one large embryo, which is allowed to develop to term.
For illustrative purposes, these results are presented by amalgamating the seven categories into three groups: affluent (categories 1 and 2), intermediate (categories 3 5) and deprived (categories 6 and 7).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com