Hi, I've often had difficulty how to use article(a/an) correctly; because of a number of nouns and their characteristics.

I'll give an example with a noun 'explanation'.

Definition 1 : [C, U] ~(for sth/ for doing sth) a statement, fact, or situation that tells you why sth happened : a reason given for sth:

-->As you see in [C, U], this noun(explanation) is counterble or noncounterble.

I want to have my question clarified with example sentences below.

connected example sentences relative to definition 1.

1. She didn't give a very adequate explanation for being late.

2. an explanation as to why he had left early.

3. She left the room abruptly without explanation.

4. to offer / provide an explanation.

I suppose the meaning of the explanation in 1 and 2 is only one (not two or more) explanation(or reason). If so, I believe, article can be omitted. (1. ~ give very adequate explanation. 2. explanation as to ~)

Then, what about the examples 3 and 4?

Does 'explanation' in the example mean more than two explanations(or reasons) or ..??

Is 'explanation' in 3 and 4 be able to used either with an or none?
Then, what about the examples 3? Does 'explanation' in the example mean more than two explanations(or reasons)-- No, it means without the uncountable process of explanation.
The other sentences need the article for the reason you gave: any single explanation.
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Thank you for your kindly answers to my questions.

I've been waiting for an answer to this question.

You gave one in the end. Thank you so much!

So you mean, since example 3 is uncountable(unprocessable-.-?) situation, article is not necessary but others are, not optional? ,,
That's right. The word is usually countable.