January has C(31,2) - 30 = 435 possibilities, of which 29 include January 31.

I'm going to assume that, for example, a schedule that has the first and 15th of every month, is the same regardless of whether it's in a leap year or an ordinary year, even though there are more days between February 15 and March 1 in a leap year than in an ordinary year. Therefore we need consider only leap years, as any schedule that applies to an ordinary year is already included in our count for leap years.  Some schedules are leap-year only: those that include February 29 and those that include both February 28 and March 1; but all the rest are common to Ordinary and Leap years.  We shouldn't count twice the ones that are common to both.

February has C(29,2) - 28 = 378 possibilities. Of these, 27 include Feb. 1 and 27 include Feb. 29; each of the 27's includes the one pair that includes both, so 53 include one or the other.

So the combined first two months have:

(435-29)*(378-27) + 29*(378-53) schedules that don't include Feb. 29. That's 151931.

(435-29)*27 + 29*26 schedules that do include Feb. 29.  That's 11716.

Rather than do the arithmetic for the entire year, a program produces the following list after each of the twelve months of the year, with the first number representing the ways that include a date on the last day of the period up through that month; the second being the number that do not include that; the last is the total of these.

 29   406   435 

 11716   151931   163647 

 4734047   66112634   70846681 

 828325802909   11568855618432   12397181421341 

 346292753994639   4663769780588355   5010062534582994 

 144945520748912187   2024389191928845672   2169334712677757859 

 62765761146906065724   876691418766200149518   939457179913106215242 

 26242035276420067961052   353420138456187685586928   379662173732607753547980 

 10983961002969204784930368   153408065547698986037570424   164392026550668190822500792 

 4591992782415740138245091808   61843619089072407601712179440   66435611871488147739957271248 

 1922040751490740544320515774384   26844282621916547258551789556064   28766323373407287802872305330448 

So by the end of December we have counted 28,766,323,373,407,287,802,872,305,330,448 or about 28.8 nonillion ways.  Years with only 28 days in February have only  26,702,087,143,838,671,766,594,258,014,464 ways. So 2,064,236,229,568,616,036,278,047,315,984 ways are available in leap years only, due to the presence of Feb. 29 or coincidence of Feb. 28 with March 1.

Yes, I know that in the following program I should have made a loop instead of repeating most of the code.

    5   LastOcc=0:LastNotOcc=1

    6   open "24dates.txt" for output as #2

   10   MLen=31:Poss=combi(MLen,2)-(MLen-1)

   20   AppFirst=MLen-2:AppLast=MLen-2

   30   AppBoth=1:AppEither=AppFirst+AppLast-AppBoth

   40   FirstOnly=AppFirst-AppBoth:LastOnly=AppLast-AppBoth

   50   NewLastNotOcc=LastNotOcc*(Poss-AppLast)+LastOcc*(Poss-AppEither)

   60   NewLastOcc=LastNotOcc*AppLast+LastOcc*LastOnly

   70   LastNotOcc=NewLastNotOcc

   80   LastOcc=NewLastOcc

   90   print LastOcc,LastNotOcc,LastOcc+LastNotOcc

   91   print #2,LastOcc,LastNotOcc,LastOcc+LastNotOcc

  110   MLen=29:Poss=combi(MLen,2)-(MLen-1)

  120   AppFirst=MLen-2:AppLast=MLen-2

  130   AppBoth=1:AppEither=AppFirst+AppLast-AppBoth

  140   FirstOnly=AppFirst-AppBoth:LastOnly=AppLast-AppBoth

  150   NewLastNotOcc=LastNotOcc*(Poss-AppLast)+LastOcc*(Poss-AppEither)

  160   NewLastOcc=LastNotOcc*AppLast+LastOcc*LastOnly

  170   LastNotOcc=NewLastNotOcc

  180   LastOcc=NewLastOcc

  190   print LastOcc,LastNotOcc,LastOcc+LastNotOcc

  191   print #2,LastOcc,LastNotOcc,LastOcc+LastNotOcc

  210   MLen=31:Poss=combi(MLen,2)-(MLen-1)

  220   AppFirst=MLen-2:AppLast=MLen-2

  230   AppBoth=1:AppEither=AppFirst+AppLast-AppBoth

  240   FirstOnly=AppFirst-AppBoth:LastOnly=AppLast-AppBoth

  250   NewLastNotOcc=LastNotOcc*(Poss-AppLast)+LastOcc*(Poss-AppEither)

  260   NewLastOcc=LastNotOcc*AppLast+LastOcc*LastOnly

  270   LastNotOcc=NewLastNotOcc

  280   LastOcc=NewLastOcc

  290   print LastOcc,LastNotOcc,LastOcc+LastNotOcc

  291   print #2,LastOcc,LastNotOcc,LastOcc+LastNotOcc

  310   MLen=30:Poss=combi(MLen,2)-(MLen-1)

  320   AppFirst=MLen-2:AppLast=MLen-2

  330   AppBoth=1:AppEither=AppFirst+AppLast-AppBoth

  340   FirstOnly=AppFirst-AppBoth:LastOnly=AppLast-AppBoth

  350   NewLastNotOcc=LastNotOcc*(Poss-AppLast)+LastOcc*(Poss-AppEither)

  360   NewLastOcc=LastNotOcc*AppLast+LastOcc*LastOnly

  370   LastNotOcc=NewLastNotOcc

  380   LastOcc=NewLastOcc

  390   print LastOcc,LastNotOcc,LastOcc+LastNotOcc

  391   print 2,LastOcc,LastNotOcc,LastOcc+LastNotOcc

  410   MLen=31:Poss=combi(MLen,2)-(MLen-1)

  420   AppFirst=MLen-2:AppLast=MLen-2

  430   AppBoth=1:AppEither=AppFirst+AppLast-AppBoth

  440   FirstOnly=AppFirst-AppBoth:LastOnly=AppLast-AppBoth

  450   NewLastNotOcc=LastNotOcc*(Poss-AppLast)+LastOcc*(Poss-AppEither)

  460   NewLastOcc=LastNotOcc*AppLast+LastOcc*LastOnly

  470   LastNotOcc=NewLastNotOcc

  480   LastOcc=NewLastOcc

  490   print LastOcc,LastNotOcc,LastOcc+LastNotOcc

  491   print #2,LastOcc,LastNotOcc,LastOcc+LastNotOcc

  510   MLen=30:Poss=combi(MLen,2)-(MLen-1)

  520   AppFirst=MLen-2:AppLast=MLen-2

  530   AppBoth=1:AppEither=AppFirst+AppLast-AppBoth

  540   FirstOnly=AppFirst-AppBoth:LastOnly=AppLast-AppBoth

  550   NewLastNotOcc=LastNotOcc*(Poss-AppLast)+LastOcc*(Poss-AppEither)

  560   NewLastOcc=LastNotOcc*AppLast+LastOcc*LastOnly

  570   LastNotOcc=NewLastNotOcc

  580   LastOcc=NewLastOcc

  590   print LastOcc,LastNotOcc,LastOcc+LastNotOcc

  591   print #2,LastOcc,LastNotOcc,LastOcc+LastNotOcc

  610   MLen=31:Poss=combi(MLen,2)-(MLen-1)

  620   AppFirst=MLen-2:AppLast=MLen-2

  630   AppBoth=1:AppEither=AppFirst+AppLast-AppBoth

  640   FirstOnly=AppFirst-AppBoth:LastOnly=AppLast-AppBoth

  650   NewLastNotOcc=LastNotOcc*(Poss-AppLast)+LastOcc*(Poss-AppEither)

  660   NewLastOcc=LastNotOcc*AppLast+LastOcc*LastOnly

  670   LastNotOcc=NewLastNotOcc

  680   LastOcc=NewLastOcc

  690   print LastOcc,LastNotOcc,LastOcc+LastNotOcc

  691   print #2,LastOcc,LastNotOcc,LastOcc+LastNotOcc

  710   MLen=31:Poss=combi(MLen,2)-(MLen-1)

  720   AppFirst=MLen-2:AppLast=MLen-2

  730   AppBoth=1:AppEither=AppFirst+AppLast-AppBoth

  740   FirstOnly=AppFirst-AppBoth:LastOnly=AppLast-AppBoth

  750   NewLastNotOcc=LastNotOcc*(Poss-AppLast)+LastOcc*(Poss-AppEither)

  760   NewLastOcc=LastNotOcc*AppLast+LastOcc*LastOnly

  770   LastNotOcc=NewLastNotOcc

  780   LastOcc=NewLastOcc

  790   print LastOcc,LastNotOcc,LastOcc+LastNotOcc

  791   print #2,LastOcc,LastNotOcc,LastOcc+LastNotOcc

  810   MLen=30:Poss=combi(MLen,2)-(MLen-1)

  820   AppFirst=MLen-2:AppLast=MLen-2

  830   AppBoth=1:AppEither=AppFirst+AppLast-AppBoth

  840   FirstOnly=AppFirst-AppBoth:LastOnly=AppLast-AppBoth

  850   NewLastNotOcc=LastNotOcc*(Poss-AppLast)+LastOcc*(Poss-AppEither)

  860   NewLastOcc=LastNotOcc*AppLast+LastOcc*LastOnly

  870   LastNotOcc=NewLastNotOcc

  880   LastOcc=NewLastOcc

  890   print LastOcc,LastNotOcc,LastOcc+LastNotOcc

  891   print #2,LastOcc,LastNotOcc,LastOcc+LastNotOcc

  910   MLen=31:Poss=combi(MLen,2)-(MLen-1)

  920   AppFirst=MLen-2:AppLast=MLen-2

  930   AppBoth=1:AppEither=AppFirst+AppLast-AppBoth

  940   FirstOnly=AppFirst-AppBoth:LastOnly=AppLast-AppBoth

  950   NewLastNotOcc=LastNotOcc*(Poss-AppLast)+LastOcc*(Poss-AppEither)

  960   NewLastOcc=LastNotOcc*AppLast+LastOcc*LastOnly

  970   LastNotOcc=NewLastNotOcc

  980   LastOcc=NewLastOcc

  990   print LastOcc,LastNotOcc,LastOcc+LastNotOcc

  991   print #2,LastOcc,LastNotOcc,LastOcc+LastNotOcc

 1010   MLen=30:Poss=combi(MLen,2)-(MLen-1)

 1020   AppFirst=MLen-2:AppLast=MLen-2

 1030   AppBoth=1:AppEither=AppFirst+AppLast-AppBoth

 1040   FirstOnly=AppFirst-AppBoth:LastOnly=AppLast-AppBoth

 1050   NewLastNotOcc=LastNotOcc*(Poss-AppLast)+LastOcc*(Poss-AppEither)

 1060   NewLastOcc=LastNotOcc*AppLast+LastOcc*LastOnly

 1070   LastNotOcc=NewLastNotOcc

 1080   LastOcc=NewLastOcc

 1090   print LastOcc,LastNotOcc,LastOcc+LastNotOcc

 1091   print #2,LastOcc,LastNotOcc,LastOcc+LastNotOcc

 1110   MLen=31:Poss=combi(MLen,2)-(MLen-1)

 1120   AppFirst=MLen-2:AppLast=MLen-2

 1130   AppBoth=1:AppEither=AppFirst+AppLast-AppBoth

 1140   FirstOnly=AppFirst-AppBoth:LastOnly=AppLast-AppBoth

 1150   NewLastNotOcc=LastNotOcc*(Poss-AppLast)+LastOcc*(Poss-AppEither)

 1160   NewLastOcc=LastNotOcc*AppLast+LastOcc*LastOnly

 1170   LastNotOcc=NewLastNotOcc

 1180   LastOcc=NewLastOcc

 1190   print LastOcc,LastNotOcc,LastOcc+LastNotOcc

 1191   print #2,LastOcc,LastNotOcc,LastOcc+LastNotOcc