1. He never failed to keep his word.

2. He never failed to keep the word. 3. He never failed to keep a word.
4. He made a word with her.
5. He had a word with her.
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I could see the pattern such as break one's word, break the word. However, I cannot see the expression, break a word. Can we use #3 or #4 containing 'a' before word? In this case, 'word' means 'promoise'.

1 and 5 are fine, but not the others. There is no substitute for either sentence.

Also, 1 and 5 have very different meanings.

In the expressions you provided, the phrase "keep one's word" means to fulfill a promise or to do what one has said they will do. It is a common expression in English.

Looking at the sentences you provided:

1. "He never failed to keep his word."
2. "He never failed to keep the word."
3. "He never failed to keep a word."
4. "He made a word with her."
5. "He had a word with her."

Let's analyze each sentence:

1. "He never failed to keep his word."
This sentence means that the person always kept their promises. The phrase "his word" refers to the promises made by the person in question.

2. "He never failed to keep the word."
This sentence is not commonly used in English. "The word" would generally refer to a specific promise or agreement that has been previously mentioned or understood in the context. Without any specific context, this sentence is unclear.

3. "He never failed to keep a word."
You are right in noticing that this expression is less common. In English, we usually use the phrase "keep one's word" without "a" before "word." This sentence, while grammatically correct, would not be commonly used. "A word" would suggest that the person never failed to keep any promise in general, rather than specific promises.

4. "He made a word with her."
This sentence is also not commonly used. The phrase "make a word" is not an idiomatic expression in English. If you want to indicate that they made a promise or agreement together, it would be better to say something like "He made a promise/agreement with her."

5. "He had a word with her."
This sentence means that the person had a conversation or discussion with her. It does not specifically refer to promises or agreements.

To summarize, the most common and idiomatic expression related to promises is "keep one's word," where "word" refers to a promise or agreement. The usage of "a" before "word" in sentences is less common, and it may not convey the intended meaning related to promises and agreements.