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For any negative integer ( − n ) :-
( − n ) ! = ( − n + 1 ) ( − n + 1 ) ! = ( − n + 1 ) ( − n + 2 ) ( − n + 2 ) ! = ( − n + 1 ) ( − n + 2 ) ( − n + 3 ) ( − n + 3 ) ! 0 0 0 0 0 0 ⋮ = ( − n + 1 ) ( − n + 2 ) ( − n + 3 ) ⋯ ( − n + n ) ( − n + n ) ! = ( − n + 1 ) ( − n + 2 ) ( − n + 3 ) ⋯ × 0 0 ! = 0 1 ( because 0! = 1 ) = undefined .
So, we have just generalized that the factorial function of any negative integer is undefined.