Recommend an Arduino Uno shield to monitor the state of 5 DPDT electrical switches
If you're really set on measuring the state of every single connection of every single switch (though I still don't see the need) then yes, you'll need 20 IO connections.
You could create those with IO expanders (MCP23017 for example), or you could use multiplexing.
Instead of 4x5 (20) IO pins you'd instead need 2+10 (12) IO pins.
Treat all the &quot;Common&quot; connections as the rows of a keypad and all the NC and NO connections as the columns of a keypad. Add a smattering of diodes to isolate everything, and you can then scan through the switches one at a time checking each of the two switch states.
You can read more about keypad matrices here.