If you meant from Pamiers and to Tarascon sur Ariège the answer is yes but there are only three buses a day, the first at 1220 and no buses at all at weekends. You would have exactly 2 hours and 50 minutes in Tarascon before the last bus back at 1610hrs.

Local bus website http://www.ariege.fr/Se-deplacer/Transports/Lignes-de-bus-regulieres

There are no buses to Carcassonne from around there. The only way is by train via Toulouse, a rather long way around.

The bus from Pamiers to Mirepoix is operated as part of the regional network and has different fares - http://cdn.ter.sncf.com/medias/PDF/midi_pyrenees/950-01-07_tcm63-38656_tcm63-43128.pdf?_ga=1.47452911.979117993.1472242214

I don't know of any bus from Pamiers to Fanjeux. Fanjeux is in a different départment and to my knowledge, other than school buses, has three buses a week from Castelnaudary and one bus a day from Carcassonne, plus another three times a week which only runs when required by people who have booked a ride.