One idea runs through these three letters: to count, and from counting, to portion out and to appoint. If you can count a set, you can divide it into shares; if you can divide it, you can assign one share to each. So the same root gives you the mathematician's quotient, the restaurant's dish, the synagogue's minyan, and the act of appointing someone to a post. This is a ל"ה root (third letter ה), which softens or drops in many forms, so the surface shapes vary while the meaning holds. Every word here is verified against pealim, niqqud included.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
The plain pa'al is the bare act of counting. The nif'al is its passive-reflexive partner: to be counted, and from there "to be counted among", to belong. The pi'el shifts the sense from counting to appointing, once you can enumerate people, you can pick one out and assign him a role. Hif'il, pu'al and huf'al are not lexicalized as everyday verbs for this root, which is normal: a root rarely fills all seven binyanim.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.