Household cleaning appliance with a single water flow path for both non-bulk and bulk dispensing

A household cleaning appliance configured to execute a cleaning cycle on an article, comprising:

• a cabinet defining an interior;
• a treating chamber located within the interior for receiving the article for cleaning;
• a dispensing system having multiple dispensing cups fluidly coupled to the treating chamber, with a first of the multiple dispensing cups defining a single use dispenser;
• a bulk dispensing cartridge received in another one of the multiple dispensing cups to define a bulk dispenser, the bulk dispensing cartridge having an outlet fluidly coupled to the first of the multiple dispensing cups through a port defined between the another one of the multiple dispensing cups and the first of the multiple dispensing cups;
• and
• a single water flow path supplying water to the dispensing system to flush treating chemistry from either of the single use dispenser or the bulk dispenser to the treating chamber.