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So, Is there any difference between these? Please correct my assumptions if I'm wrong.

I have some errands to run. "To run" here is Infinitive which acts as attributive and is used for pointing an a purpose of the item.

I have some errands to be run. The same explanations could be applied, couldn't they? And Passive form is used because the task is performed by someone else, isn't it?

Thanks in advance. Best wishes.
E.P.F.
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FandorinPassive form is used because the task is performed by someone else, isn't it?
Yes. That's the general idea.

I have some Xs to Y. I have to Y the Xs.
I have some Xs to be Yen. Somebody has to Y the Xs.

I have some letters [to write / to be written].
I have some packages [to wrap / to be wrapped].
I have some chores [to do / to be done].

(The passive form is hardly ever used in this way unless you're the boss giving commands to your employees, or something of that nature.)

My comments here are not the last word on the subject. There are many patterns with an infinitive after a noun, and each may have multiple meanings.

CJ
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Thank you CalifJim. I wish you Happy New Year and everything you want to come true! Emotion: smile
FandorinHappy New Year
And to you, too! Emotion: wink