A073531 Number of n-digit positive integers with all digits distinct.

9, 81, 648, 4536, 27216, 136080, 544320, 1632960, 3265920, 3265920

Offset: 1

Author

Zak Seidov, Aug 29 2002

Keywords

• nonn
• base
• fini
• full

Comments

For any base b the number of distinct-digit numbers is finite. For base 10, the maximal distinct-digit number is 9876543210; for any larger number at least two digits coincide. The number of distinct-digit primes is also finite, see A073532.

If "positive" is replaced by "nonnegative" we get the sequence 10, 81, 648, 4536, 27216, 136080, 544320, 1632960, 3265920, 3265920.

Alternatively, if 0 is considered to have 0 digits, one could prefix a(0) = 1. This would be compatible with the given formula and 9/10 rounded to the nearest integer. - M. F. Hasler, Dec 10 2018

a(10) is the final term because no number having more than 10 digits can have all digits distinct. - Jon E. Schoenfield, May 17 2021

Examples

			a(3) = 648 because there are 648 three-digit integers with distinct digits.
		

Links

• Eric Weisstein's World of Mathematics, Digit

Crossrefs

Cf. A073532.

Cf. A010784 for the list of these integers.

Programs

GAP

List([1..10],n->9*Factorial(9)/(Factorial(10-n))); # Muniru A Asiru, Dec 11 2018

Magma

[9*Factorial(9)/Factorial(10-n): n in [1..10]]; // Vincenzo Librandi, Dec 13 2018

Maple

seq(9*factorial(9)/(factorial(10-n)),n=1..10); # Muniru A Asiru, Dec 11 2018

Mathematica

Table[9*9!/(10-n)!, {n, 10}]

PARI

apply( A073531(n)=if(n<11,9*9!\/(10-n)!), [1..13]) \\ or: 9*binomial(9,10-n)*(n-1)! without need for if(). - M. F. Hasler, Dec 10 2018

Formula

a(n) = 9*9!/(10-n)!.

Extensions

Keywords fini, full added by Jon E. Schoenfield, May 17 2021