Serial Number Combinations

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I am hoping that we have a maths nerd out there (I failed my 11+ ).

I am trying to ascertain the following:
How many times would 4444 occur in a run of 10M?
How many times would 55555 occur in a run of 10M?
How many times would 666666 occur in a run of 10M?

Many thanks

I think you need to look at how many ways each of those can occur.

For example 4444 could be 4444ABC OR A4444BC OR AB4444C OR ABC4444.

That's four different possibilities.  For each of these, ABC could be any three digits, more or less (depending on how you want to count the duplicate results).  Let's just say the ABC could be any number(s) from 0-9.  That gives us 10 possibilities for each so 10 * 10 * 10 * 4 or 4000 different ways to get 4444.  This does count some notes more than once such as 4444444 being counted four times so you might make make some adjustments for those cases but that should put you in the ballpark.

I put this into chatgpt & clarified a 7 digit possibility with 4444. It said there were 3,700 possibilities and created this spreadsheet.

Here are the exact counts and downloadable CSVs you asked for.

"55555" (length 5) — 280 seven-digit numbers contain this substring.
Download numbers_with_55555.csv

"666666" (length 6) — 19 seven-digit numbers contain this substring.
Download numbers_with_666666.csv

For reference, the probabilities (out of the 10,000,000 possible 7-digit strings) are:

"55555": 280 / 10,000,000 = 0.000028 (≈ 0.0028%)

"666666": 19 / 10,000,000 = 0.0000019 (≈ 0.00019%)

I plugged this question into "HI" instead of AI and it took 30 seconds to get an answer.  My answer is the same as AL-Bob's post.  The logic applied is simple using 0-9 in each digit and deducting the count for duplicate 7 digit solid numbers.

(HI: human intelligence )