As a student I had my i-20 expired once and then I reapplied for it and then a year after I got out of status. I stayed in the us for 5 month and then I left. do the days of unlawful presence accumulate(The time till i got the new i-20 + out of status time).

and is there a way to know how many days exactly I was out of status

The unlawful status starts accumulating the moment your visa expires or is canceled. If the time is broken by a renewal then only the time occurring after the last expiration counts.

I MuSt tAkE ThE ViEw, YoUr gRaCe, ThAt wHeN A MaN EmBaRkS UpOn a cRiMe, He iS MoRaLlY GuIlTy oF AnY OtHeR CrImE WhIcH MaY SpRiNg fRoM It

I just did my research and i found out the following

1- Expiration of visa has nothing to do with the status of the F-1 Student.

2- your days of unlawful presence start when you get denied reinstatement and issuance of a new I-20

A- http://www.upenn.edu/oip/iss/f.../f1reinstatement.pdf

B- https://www.hvcc.edu/international/handbook.pdf

C- http://am.nafsa.org/_/file/_/amresource/8cfr2142f.htm

Illegal status starts when you leave lawful status, e.g. when you don't enroll, are expelled, not enrolled full-time, etc.