"AND" is not really possible in regexes. You have to use advanced tricks such as positive lookaheads.

/^(?=.*[A-Za-z])(?=.*[0-9])[A-Za-z0-9]{3,10}\z/

[download]

Of course, you could possibly use multiple regexs for increased readability.

/^[A-Za-z0-9]{3,10}\z/ && /[A-Za-z]/ && /[0-9]/

[download]

Update: For fun,

Eliminates one lookahead: 

/^(?=[A-Za-z0-9]{3,10}\z).*(?:[0-9][A-Za-z]|[A-Za-z][0-9])/

[download]

Eliminates both lookahead: 

/(?>(?:[0-9][A-Za-z]|[A-Za-z][0-9]).*\z)(?(?{pos()<3||pos()>10})(?!))/

[download]

All tested.

Update: Benchmarked. (Benchmarking Code, a prime example of a good Benchmark if you ask me.) 

----------------------------------------
Short (arg: 2j)

            Rate ikegami4 ikegami3 ikegami2 ikegami1
ikegami4  573/ms       --     -60%     -76%     -77%
ikegami3 1428/ms     149%       --     -40%     -44%
ikegami2 2392/ms     318%      68%       --      -5%
ikegami1 2529/ms     342%      77%       6%       --

----------------------------------------
Fine (arg: n4a1z9)

           Rate ikegami4 ikegami2 ikegami1 ikegami3
ikegami4 638/ms       --     -11%     -18%     -20%
ikegami2 715/ms      12%       --      -8%     -10%
ikegami1 776/ms      22%       9%       --      -2%
ikegami3 794/ms      24%      11%       2%       --

----------------------------------------
Just Letters (arg: nhfdmzaa)

           Rate ikegami4 ikegami3 ikegami2 ikegami1
ikegami4 397/ms       --     -16%     -44%     -45%
ikegami3 471/ms      19%       --     -34%     -35%
ikegami2 714/ms      80%      52%       --      -2%
ikegami1 728/ms      83%      55%       2%       --

----------------------------------------
Long (arg: qq3m3aaaazz3)

            Rate ikegami4 ikegami1 ikegami3 ikegami2
ikegami4  157/ms       --     -79%     -88%     -89%
ikegami1  746/ms     376%       --     -45%     -45%
ikegami3 1351/ms     762%      81%       --      -1%
ikegami2 1365/ms     770%      83%       1%       --

[download]

My first choice based solely on benchmarks would be ikegami2 (the three regexps).
My first choice based solely on benchmarks would be ikegami1 (straightforward lookahead).

In reply to Re: A hard RegEx problem. by ikegami
in thread A hard RegEx problem. by pysome

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.