Thinking About Thinking

How Many Unique Combinations Are There?

Posted in Pop Culture, Technology by larrycheng on August 16, 2010

I was thinking randomly about how many combinations there are and how much unique shortened URLs they can generate given their current construct.  Here’s their construct:

  • Up to 6 characters after
  • Any mix of capitalized letters, lowercase letters, and numbers

If you limit the combinations to only ones where is the domain (and not partner domains), and you presume they can go up to 6 characters (i.e. 1–6) as opposed to exactly 6 characters – how many total combinations are there?  Any math whiz out there who can figure it out pretty quickly and enlighten us in the comments section?  My small math brain was getting cramps thinking about it. 

Given that they shorten 40–50 million URLS per day, I also wonder how long they can last on this construct.  Probably a long time I suspect even including their growth.  Random thought for the day, and thanks for the help. 


30 Responses

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  1. Charlie Kroll said, on August 16, 2010 at 9:22 am


    • larrycheng said, on August 16, 2010 at 9:23 am

      Charlie – how’d you get to that answer? Sounds plausible.

  2. Charlie Kroll said, on August 16, 2010 at 9:29 am

    Assumptions were…
    * 62 possible characters (26 letters, each can be upper or lower case, plus 10 numerical characters).
    * String can be 1-6 characters long

    So, calculation would be (62^6)+(62^5)+(62^4)+(62^3)+(62^2)+(62^1)

    Make sense?

    • Dave Gordon said, on August 16, 2010 at 10:12 am

      I think the answer is actually slightly different. This is a permutations problem requiring the formula n!/(n-k)! where “n” is the set of values you are choosing from (in this case the number of possible characters) and “k” is the length of the string. Using 62 possible characters and strings of 6, 5, 4, 3, 2, 1 your formula would be:

      62!/(56!) + 62!/(57!) + 62!/(58!) + 62!/(59!) + 62!/(60!) + 62!/(51!) =

      45,051,792,964 possible permutations

      • Dave Gordon said, on August 16, 2010 at 10:15 am

        actually nevermind, this would assume the use of each value only once, which in a url isnt a requirement – so charlie’s answer is right. in any case, now you know how to solve for permutations!

  3. said, on August 16, 2010 at 10:28 am

    Thanks for asking! I have always wondered the same thing, and keep waiting for the day they extend to 7 characters.

    FYI: All links use the same shortened URL regardless of “partner domains”

    The shortened link for this post is:

    A post from TechCrunch this am is:

    However, you can’t hijack someone else’s shortened URL and try to make it your own:

    Another consideration is custom domains longer than 6 characters like:

  4. Hemang said, on August 16, 2010 at 10:32 am

    I do believe Charlie has the right answer. The breakout is as follows:

    62^6= 56800235584
    62^5= 916132832
    62^4= 14776336
    62^3= 238328
    62^2= 3844
    62^1= 62

    Total = 57,731,386,986

    • Hemang said, on August 16, 2010 at 10:35 am

      Just to clarify Larry, 62^6 represents the number of combos if you could ONLY have 6 characters. So you take the total combos for each number of characters possible and that should be the right number.

  5. larrycheng said, on August 16, 2010 at 10:17 pm

    Thanks all – sounds like we have a consensus!

  6. Desmond Pieri said, on August 18, 2010 at 5:48 pm

    Thanks for figuring this out guys. I wasn’t able to sleep nights recently, worrying if I was using up more than my fair share of bit.lys. Now I can rest assured. Des

  7. Julien Nakache said, on August 20, 2010 at 11:27 am

    The math behind that are so trivial that I don’t understand the point of this article. Was that a way to get some comments on your blog ?

    Idea for a next blog post : now that we know how many different urls can generate and the fact they are shortening 50million links a day, how long will they be able to shorten urls ? Seems complicated…

    Sorry for being sarcastic, but seriously, it s VERY basic math…

    • larrycheng said, on October 5, 2010 at 4:38 pm

      Julien and Marco – I guess I’m just a mere math mortal. I can live with that!

  8. Marco Gomes said, on August 24, 2010 at 11:49 pm

    I am brazilian, brazilian education is made for monkeys, and even I knew the answer for this math question 🙂

    (I am brazilian, I can make jokes with brazilian educational system 🙂

  9. Dave Rodecker said, on September 28, 2010 at 1:07 am

    Extra credit assignment:
    If in the new land of IPv6, there exists a need to create unique IPs for each URL.
    So, if the entire scope of about 62^6 URLs were mapped to IPv6 addresses, what is the minimum IPv6 subnet that could contain it?

  10. Athman Mohamed said, on December 25, 2010 at 3:09 pm

    Interesting thought. I had another one, I once typed in into and it gave me this:

    is THAT shortening? LoL 😀

  11. Andrew said, on January 3, 2011 at 6:19 pm

    26 letters, upper and lower case, plus 0-9 = 72 unique characters, right?

    72^6 = 139,314,069,504

  12. J.S. said, on January 5, 2011 at 6:09 am

    26 + 26 + 10 = 62

    I wondered that bit ly’s don’t give URL’s that vary from 1 to 6 strings; they are always exactly 6 strings
    So, the number should be
    62 ^ 6 = 56, 800, 235, 584
    Correct me if im wrong, but I never saw a bit ly less than 6 characters.

  13. barry said, on March 23, 2011 at 6:23 am

    Which surely means will run out in about 3 years. (50 thousand million @ 50 million per day = 1000 days = 3 years all approximations).So time to add an extra character?

  14. Drew said, on March 28, 2011 at 9:26 pm

    If they upped it to 8 characters then it would be 62^8 = 2.18340106 × 10^14

    Considering IPv4 allows 32 bits for an Internet Protocol address, and can therefore support 2^32 (4,294,967,296) addresses, IPv6 uses 128-bit addresses, so the new address space supports 2^128 (approximately 340 undecillion or 3.4×10^38) addresses.

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  20. Sam Woods said, on March 26, 2014 at 5:54 am

    That’s a lot of possible shortened links! must be highly optimized to handle that. They’re also adding other small domains arn’t they?

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