While studying yesterday, I came across the following sentence: "バスの中で お客さんの数を
数えました" (the spaces are included in the exercise). The meaning is simple
enough, but when I asked my teacher whether that "の数" is necessary for the
sentence to be correct, she said yes. I still don't quite understand, however,
what kind of things you could use as a direct object with "数える". What exactly
can be "counted"? I've seen at least one example where the word "人数" was used.
Can you use any noun that doesn't end in "数" as the direct object (that is, as
the receiver of the 'を' particle) with the verb?

Edit: I was looking at examples of using this verb on the JLearn website; And
I think I might understand better what is going on. Although I would
appreciate very much if someone with better knowledge of the language could
explain it.

On the website, there are a few examples that use the 'を' particle with the
数える verb. Most of these have as direct object either the noun '数', being
further specified by a 'の' like in the example I made, or some other word that
ends in that kanji, such as 人数. One used what a similar word, '量'. Most of the
others, however, seem to use a different construction; counting the object
inside some class. Two examples of these are:

"君は私を君の友人のうちに数えてよい" "You can number me among your friends"

"彼は世界で最も偉大な科学者の中の一人に数えられている" "He is numbered among the greatest scientists in
the world"

All of these examples seem to further use an indirect object to tell us what
the direct object is being counted as. As A. Ellett mentioned in the comments,
this isn't really counting; but an expression that sounds similar in English.

My suspicion is that the issue with not using the word '数', or some other
similar one with this verb is that doing so would imply this second
construction. There is one example that doesn't fit this, however:

"彼は薄暗がりと戦いながら、彼等の名前を数えていった" "Fighting the fading light he continued to count
their names"

I am not sure why this sentence in particular works; but it was the only
example where I found a noun that is not a number being counted without a "に"
particle to indicate what it is being counted as.
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question
verbs particle-を
How to use the verb "数える" properly?
This is not a clear answer, but hopefully helps.

First, to me, バスの中でお客さんを数えた does not sound completely wrong. It is less
natural, but I would say it is acceptable.

A possible explanation is that 数える can take a specific set of things as an
object. So

  * 本を数えた I counted books

sounds rather strange, but

  * 本棚の本を数えた I counted the books on the shelf.

sounds OK.

I guess this is similar in English: _I counted books_ should be strange while
_I counted **the** books_ **might** be acceptable. But Japanese does not have
articles, restricting the _books_ by a modifying phrase is the only
possibility.

Back to your example, お客さんを数えた sounds strange, but having バスの中で, it is clear
that you are talking about the passengers on the bus, so the sentence is
acceptable.

**Other examples**

  * 電車でスマホを見ている人を数えた I counted the number of people watching smartphones on the train

  * 締め切りまでの日にちを数えた I counted the number of days before the deadline.

  * テーブルの上の空き瓶を数えた I counted the number of empty bottles on the table.

  * 残りのビールを数えた I counted the number of (cans or bottles of) beer left.

===

The only noun I came up with that can be used for 数える without any -数 noun or
any modification is money: お金を数える

Since this implies you are counting coins and notes you have, it may not be
really an exception.
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answer