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How many different combinations can quarks produce? E.g. 2 up quarks + 1 down quark = proton. Is this value going to be infinite?

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There's 2 types of particles that quarks can form, Baryons and Mesons.

List of Baryons and List of Mesons

All of these, except for the Proton are unstable. The Neutron is unstable on it's own but it's stable when bound to Protons in an atomic Nucleus.

A Baryon is 3 quarks and a Meson is 2 quarks, so that limits the number of combinations.

Description here

Baryons need 3 quarks to maintain color neutrality (1 red, 1 blue, 1 green) and with 6 types of quarks, that's 216 possible combinations, but the top quark doesn't form any baryons, so that leaves 75 possible Baryons - which is the number listed in the list of Baryons. Now all 75 might not exist, but that's the maximum number.

Top Quarks not expected to form Baryons explanation here

The meson list (linked above) is more confusing to me. Assuming the same rule for top quarks applies, which seems to be the case explanation here, There should be 5 possible quarks and 5 possible anti quarks, making 25 possible Mesons, I would think, but the list has 64 possible mesons including separate tables for Pseudoscalar mesons and Vector mesons - so, . . . not sure what's going on there.

That's as far as I can answer it, anyway, if anyone wants to give a more detailed answer, or correct errors, feel free.

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    $\begingroup$ The reason there are more than 25 mesons is that a single quark combination (like $u\bar{d}$) can be any of several mesons, depending on the quarks' angular momentum and quantum state. If you check carefully you'll notice that there are 25 separate pseudoscalar mesons (a couple of the K's are double-counted) and 25 vector mesons. $\endgroup$ – David Z Jul 13 '15 at 6:11
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All possible baryons "http://members.quicknet.nl/nj_dewit/quarks/p6B.htm".

All possible mesons "http://members.quicknet.nl/nj_dewit/quarks/p6M.htm".

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