Five digit numbers containing all digits from 1-5 are arranged from highest to lowest. What is the 25th number?
(Examples of these numbers are 54 321; 23 451; 41 532)
My Solution:
First, it’s impractical to list these (5)(4)(3)(2)(1)=120 numbers from highest to lowest and then determine the 25th number in the list. So, we can instead do some technique and investigate the distribution of these numbers.
Since it’s from highest to lowest, the first step is to determine the number of these five digit numbers which have a ten thousands digit of 5.
For thousands digit: 4 numbers to choose.
For hundreds digit: 3 numbers to choose.
For tens digit: 2 numbers to choose.
For unit digit: 1 number to choose.
Thus, the number of these 5 digit numbers which have a ten thousands digit of 5 is (4)(3)(2)(1)=24. Then these 24 numbers are the first 24 numbers in the list. Hence, the 25th number is the highest number which has a ten thousands digit of 4. And this 25th number is 45 321.
I like your solution, but I do not agree that listing 25 5-digit numbers is impractical (and we don’t need to list the other 85)
Jonathan