  Ca-rich transients (CRTs, 2005E-like supernovae) exhibit unusually strong Ca
features during their nebular phase, and their distribution in their host
galaxies indicates that they belong to a metal-poor old population.
  We carried out a series of binary population synthesis and presented four
different channels of forming CO WD + He WD pairs. We selected the systems
fulfilling the constraints of the old population and the birth rate from all
the CO WD + He WD pairs by constraining the component mass of the WD pairs.
  For the four channels, the stable Roche lobe overflow (RLOF) could have a
significant influence on the formation of the WD + WD pairs. Based on their
position on the $M_{\rm CO}$-$M_{\rm He}$ plane, the mass-transfer between the
components for most of the CO WD + He WD pairs is neither always unstable nor
always stable. We found that it is necessary that the CO WDs are less massive
than 0.6 $M_{\odot}$ and the He WDs are less massive than 0.25 $M_{\odot}$ if
CO WD + He WD pairs fulfill the constraints from both the old population and
the birth rate of CRTs. However, the He WD mass is lower than, while the total
mass of the low-mass WD pairs is larger than{\bf ,} the ejecta mass of the CRTs
derived from observations.
  Our results imply that the CO WDs participate in CRT explosions and at the
same time, a bound remnant could be left after the CRT explosion if the
low-mass WD pairs are the progenitors of CRTs. Therefore, whether or not the
helium detonation on a low-mass CO WD may lead to the second detonation in the
center of the CO WDs, and whether or not the binding remnant is left after the
thermonuclear explosion should be examined carefully in the future.
