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 Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 800 characters in body Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 13 characters in body Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? deleted 2716 characters in body Feb 23 answered How many different numbers can be written if each used digit symbol is used at least 2 times? Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 124 characters in body Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 124 characters in body Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 124 characters in body Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 124 characters in body Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 124 characters in body Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? edited body Feb 23 comment How many different numbers can be written if each used digit symbol is used at least 2 times? @MarkusScheuer : Thank you a lot for your answer. It gave me a great point how to find the general formula of $P(n,d)$. Now I am testing it for some values. Best Regards Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 611 characters in body Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? edited tags Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 51 characters in body Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 10 characters in body Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 191 characters in body Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? deleted 2 characters in body Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 674 characters in body Feb 23 comment How many different numbers can be written if each used digit symbol is used at least 2 times? @MarkusScheuner : And then easily $A_n(x)=(e^x - x)^n$ can be gotten. It is very nice result Feb 23 comment How many different numbers can be written if each used digit symbol is used at least 2 times? @MarkusScheuner : Wonderful relations . Please check the relation while applying the recurrence relation in (4) . I think it should be $$\sum_{d=0}^\infty (P(n+1,d)+d.P(n,d-1))\frac{x^d}{d!}\\$$. then we can get $$A_{n+1}(x)=(e^x-x)A_{n}(x)$$