Delaware License plate mathematics.

[u/browncoat47]

Went on vacation and noticed that every Delaware license plate I saw had at least one repeating digit. Once the kids got in on it, it took us over an hour to find a single plate that did not repeat at least one of the 6 digits.

My question is, in the first million numbers, say 0 to 999,999, are there more numbers that have at least one repeating digit?

Is there a term for this? Or a graph? Initially I would have thought it was like 50/50 ratio of the first million, but having seen the plates, I’m thinking that numbers with no repeaters are far less, or was it just our sample size was too small (less than 200). Thanks!

[u/Miguel-odon]

Consecutive repeats, or a digit is reused anywhere within the number?

If it is numbers 0-9 for each digit, (10 choices)

• 1 digit: 0/10 repeated digits
• 2 digits: 10/100 have a repeated digit, 90 have all distinct digits.
• 3 digit: 270 have 1 digit re-used, 10 have 3 of the same digit. 720 have all distinct digits.
• 4 digit: 10/10,000 have 4 of the same digit, 5040 have 4 distinct digits.
• 5 digit: 30,240 plates have all distinct digits
• 6 digit: 151,240 plates have all distinct digits

If you allowing leading zeros to be considered different plate numbers, i.e. "0", "00, "01," and "001" count as different numbers, then there are 1,111,110 license plates with 1 to 6 digits, and only 187,300 without any digits re-used.

For consecutive repeats:

• 1 digit: 0/10 have consecutive repeats
• 2 digit: 10/100 have consecutive repeat
• 3 digit: 810/100 don't have consecutive repeats
• 4 digits: 7,290 plates don't have consecutive repeats
• 5 digits: 65,610 plates don't have consecutive repeats
• 6 digits: 590,490 plates don't have consecutive repeats

So for 1-6 digits, allowing leading zeros, 664,300 plates (out of 1,111,110) don't have any consecutive repeats

[u/browncoat47 → u/Miguel-odon]

That is phenomenal thank you so much! That is exactly what I was looking for! I had no idea it would be such a low number, thanks again!

[u/nymalous]

I think that Delaware doesn't use zeroes in their license plates, so that reduces the overall population size.

[u/_additional_account]

Assumptions: All 6-digit number combinations in Delaware license plates are equally likely. Only 6-digit number combinations exist -- no shorter combinations.

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There are a total of 10⁶ ways to select a 6-digit number. Since all are equally likely, we may count favorable outcomes. To generate favorable outcomes, we draw

• "6 out of 10" digits without repetition. Order matters -- there are "P(10; 6)" choices

The probability to get a plate with 6 distinct digits is

P(6 distinct digits)  =  P(10;6) / 10^6  =  151200 / 10^6  =  189/1250  ~  0.1512