Xenodrome Probabilites

I posted about xenodromes in an eponymous post on the 20th October 2024. In this current post, I want to look at the probabilities of xenodromes occurring in the various number bases. Let's start with base 16 where we have 16 digits to choose from:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f

An example of a five digit xenodrome in base 16 would be 56a3f where:$$56a3f_{16}=354879_{10}$$What I found was the following (permalink):

There are 491400 permutations of 5 digits in base 16 so that all are distinct with no leading zero. There are a total of 983040 possible permutations of 5 digits in base 16 with no leading zero. Probability of a 5 digit xenodrome in base 16 is thus 0.4999 or 50%.

The probability is thus effectively 50%. However, the five digit decimal numbers that I deal with when investigating the numbers associated with my diurnal age (10000 to 40000) require only four hexadecimal digits for their representation and so:

There are 40950 permutations of 4 digits in base 16 so that all are distinct with no leading zero. There are a total of 61440 possible permutations of 4 digits in base 16 with no leading zero. Probability of a 4 digit xenodrome in base 16 is thus 0.6665 or 67%.

So 5 digit base 10 numbers in the range between 10000 and 40000 will have a 2/3 chance of being xenodromic in base 16. 

Let's look at the other bases now. Here's the results for base 15 and 5 digits:

There are 336336 permutations of 5 digits in base 15 so that all are distinct with no leading zero. There are a total of 708750 possible permutations of 5 digits in base 15 with no leading zero. Probability of a 5 digit xenodrome in base 15 is thus 0.4745 or 47%.

 For base 14 and 5 digits:

There are 223080 permutations of 5 digits in base 14 so that all are distinct with no leading zero. There are a total of 499408 possible permutations of 5 digits in base 14 with no leading zero. Probability of a 5 digit xenodrome in base 14 is thus 0.4467 or 45%.

For base 13 and 5 digits:

There are 142560 permutations of 5 digits in base 13 so that all are distinct with no leading zero. There are a total of 342732 possible permutations of 5 digits in base 13 with no leading zero. Probability of a 5 digit xenodrome in base 13 is thus 0.4160 or 42%.

For base 12 and 5 digits: 

There are 87120 permutations of 5 digits in base 12 so that all are distinct with no leading zero There are a total of 228096 possible permutations of 5 digits in base 12 with no leading zero Probability of a 5 digit xenodrome in base 12 is thus 0.3819 or 38%.

For base 11 and 5 digits: 

There are 50400 permutations of 5 digits in base 11 so that all are distinct with no leading zero There are a total of 146410 possible permutations of 5 digits in base 11 with no leading zero Probability of a 5 digit xenodrome in base 11 is thus 0.3442 or 34%.

For base 10 and 5 digits:

There are 27216 permutations of 5 digits in base 10 so that all are distinct with no leading zero There are a total of 90000 possible permutations of 5 digits in base 10 with no leading zero Probability of a 5 digit xenodrome in base 10 is thus 0.3024 or 30%.

So the probability of a five digit xenodrome in base 10 is 30%. We may as well continue for the lower bases.

For base 9 and 5 digits:

There are 13440 permutations of 5 digits in base 9 so that all are distinct with no leading zero. There are a total of 52488 possible permutations of 5 digits in base 9 with no leading zero. Probability of a 5 digit xenodrome in base 9 is thus 0.2561 or 26%.

For base 8 and 5 digits: 

 There are 5880 permutations of 5 digits in base 8 so that all are distinct with no leading zero. There are a total of 28672 possible permutations of 5 digits in base 8 with no leading zero. Probability of a 5 digit xenodrome in base 8 is thus 0.2051 or 21%.

For base 7 and 5 digits:

There are 2160 permutations of 5 digits in base 7 so that all are distinct with no leading zero. There are a total of 14406 possible permutations of 5 digits in base 7 with no leading zero. Probability of a 5 digit xenodrome in base 7 is thus 0.1499 or 15%. 

For base 6 and 5 digits:

There are 600 permutations of 5 digits in base 6 so that all are distinct with no leading zero. There are a total of 6480 possible permutations of 5 digits in base 6 with no leading zero. Probability of a 5 digit xenodrome in base 6 is thus 0.09259 or 9%. 

For base 5 and 5 digits:

There are 96 permutations of 5 digits in base 5 so that all are distinct with no leading zero. There are a total of 2500 possible permutations of 5 digits in base 5 with no leading zero. Probability of a 5 digit xenodrome in base 5 is thus 0.03840 or 4%.

Of course, there are no 5 digit xenodromes in base 4 so we need to switch to four digit numbers. So for base 4 and 4 digits:

There are 18 permutations of 4 digits in base 4 so that all are distinct with no leading zero. There are a total of 192 possible permutations of 4 digits in base 4 with no leading zero. Probability of a 4 digit xenodrome in base 4 is thus 0.09375 or 9%.

Similarly there no 4 digit xenodromes in base 3 so we need to switch to three digits. For base 3 and 3 digits:

There are 4 permutations of 3 digits in base 3 so that all are distinct with no leading zero. There are a total of 18 possible permutations of 3 digits in base 3 with no leading zero. Probability of a 3 digit xenodrome in base 3 is thus 0.2222 or 22%.

Finally, for base 2 and 2 digits we have:

There is 1 permutation of 2 digits in base 2 so that all are distinct with no leading zero. There are a total of 2 possible permutations of 2 digits in base 2 with no leading zero. Probability of a 2 digit xenodrome in base 2 is thus 0.5000 or 50%.

Here is the permalink again to carry out these calculations. Just adjust for the number of digits and the base.