"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.)

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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%       --

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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%       --

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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%       --

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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).

Is there a specific reason why you paired ^ with \z? Normally, I'm used to seeing either ^/$ (start/end of line) or \A/\z (start/end of string). Mixing those up to anchor one end of the line and the other end of the string seems strange to me.

Happenstance. I chose them independently, not as a pair. Use \A..\z if you prefer. \z is kinda odd anyway. \A is closer to \Z than to \z.

Agreeed, but then they wouldn't be solving the homework problem as it was given . . .