I am wondering how 組まされた was conjugated.
I know it is a form of the verbs 組む and する, but I don't know why む became ま, instead of the stem plus the passive form of する.
Is there something I am not getting?
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This is actually a causative-passive construction.
Sometimes there is confusion around causative forms because there are actually two causative verbal forms for many verbs. For godan verbs like 組む, there is a 'standard' causative form in which you attach せる to the a-row (未然形) of the verb. For 組む this would be 組ませる. There is also a 'short' causative form in which you attach す to the a-row (未然形) of the verb. For 組む this would be 組ます. To then make the causative-passive for each of these forms, you get せられる for standard forms and される for short forms.
組む－組ませる ('standard' causative)－組ませられる ('standard' causative-passive)
組む－組ます ('short' causative)－組まされる ('short' causative-passive)
So 組まされた is the past tense of the 'short' form causative-passive.
In general, the 'short' forms are used in informal conversation and the 'standard' forms are used both in written and spoken Japanese. For example, ichidan verbs like 食べる have two causative forms (食べさせる and 食べさす). You might have heard the phrase 食べさして in informal Japanese, the short-causative form (食べさせて is the 'standard' form). Such a form would never be used in a formal situation.
Check out Tae Kim's guide on these verbal forms here.