DAX basis different date columns

Try

Num items =
VAR SelectedMonth =
    MAX ( 'Date'[Date] )
VAR StartCurrentMonth =
    EOMONTH ( SelectedMonth, -1 ) + 1
VAR MidPrevMonth =
    EOMONTH ( SelectedMonth, -2 ) + 16
VAR Result =
    CALCULATE (
        COUNTROWS ( 'Table' ),
        'Table'[Expiration Date] < MidPrevMonth,
        'Table'[Received Date] < MidPrevMonth,
        'Table'[Billed Date] > StartCurrentMonth
            || ISBLANK ( 'Table'[Billed Date] ),
        REMOVEFILTERS ( 'Date' )
    )
RETURN
    Result

The REMOVEFILTERS is strictly only necessary if you have a relationship from your date table to the fact table.

Also, this assumes that there is one row per item. If that isn't the case then you can replace COUNTROWS with a DISTINCTCOUNT of the id column.