WHY IS ZERO FACTORIAL EQUALS 1? WHY FACTORIAL OF NEGATIVE NUMBER (-1) DOES NOT DEFINE????

GENERAL FORMULA OF FACTORIAL:-

(N+1)! = (N+1)×N!

PROOF WHY ZERO FACTORIAL = 1

By the formula,(N+1)!=(N+1)×N!

        5!=5×4!

          4!=4×3!

         3!=3×2!

         2!=2×1!

         1!=1×0!

  We know 1! =1 then,

         1=1×0!

        1/1=0!

      THUS ,,,,,,,,,0!=1

FACTORIAL OF NEGATIVE NUMBER    (-1)     DOESN'T DEFINE WHY?

AGAIN FROM THE FORMULA 

          (N+1)!=(N+1)×N!

   

     2!=2×1!

     1!=1×0!

      0!=0×-1!   ...................  ?????????,

HERE CONTRADICTION IS ARISE!!!!

        0!=0×-1!

    0!÷0 = -1! ,,,,,

           But,,,,        0! ÷0    IS    UNDEFINE .......

So FACTORIAL of  negative number    (-1)        DOESN'T exist.....

IF SOMEWHERE IS WRONG 

PLEASE NOTIFY ME..........

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