The Descending Integers (Posted on 2006-02-02)

	Submitted by K Sengupta
Rating: 2.8000 (5 votes)
Solution:	(Hide)
The required number is 9,403,152,678. EXPLANATION: At the outset, the total number of integers commencing with 9 and possessing all the nine digits is 9! = 362,880. Since, 200,136 is less than 362,880; it follows that the first digit must be equivalent to 9. Since, all the digits are non-repetitive, the value of the second digit is 8 or less. Now, 200,136 = 4*(8!) + 38,856 and accordingly, the second digit must be 8-4=4. 38,856 = 7*(7!) + 3,576. since the third digit must contain one of 8,7,6,5,3,2,1,0 ; it follows that the said digit must correspond to 0. 3,576 = 4*(6!) + 696. Since, the fourth digit must contain one of 8,7,6,5,3,2,1 it follows that the said digit must correspond to 3. 696 = 5*(5!) + 96. Since, the fifth digit must contain one of 8,7,6,5,2,1 ; it follows that the said digit must correspond to 1. 96 = 3*(4!) + 24. Sice the sixth digit must contain one of 8,7,6,5,2 ; it follows that the said digit must be equal to 5. Now, 24 = 4!. Accordingly, the digits of the smallest four digit number constituted by 8,7,6,2 must correspond to the last four digits and consequently, the last four digits ( written from left to right) are 2,6,7,8 giving the required number as 9,403,152,678. ---------Q E D----------- COMPUTER SOLUTION (Submitted By Charlie): By taking the permutations of all ten digits in ascending order of value, and counting down from 10! = 3628800, when we get to a count of 200136, that will be the 200,136th in descending order. DECLARE SUB permute (a$) DEFLNG A-Z CLS a$ = "0123456789": h$ = a$ ct = 3628800 DO permute a$ ct = ct - 1 LOOP UNTIL a$ = h$ OR ct = 200136 PRINT a$ The permute subroutine is shown elsewhere on this site. The answer agrees, as 9403152678.

	Subject	Author	Date
	Acknowledgements	K Sengupta	2006-03-15 01:13:34
	Correct Solution by hand	Bill Matthews	2006-02-04 07:56:17
	I see my error...	Dej Mar	2006-02-03 01:25:39
	My calculations	Dej Mar	2006-02-03 01:23:22
	If my calculations have been correct,,,,	Dej Mar	2006-02-02 21:32:34
	re: solution -- computer confirmation	Charlie	2006-02-02 12:45:28
	re: solution I agree	goFish	2006-02-02 12:38:48
	solution	Charlie	2006-02-02 12:23:32