by +ing點用咩意思

"sleep-deprived drivers"-----subject
(have fallen asleep)-------verb in predicate
and they have been falling asleep-----present perfect continuous tense, another subject
by falling asleep------a phrase as a group of words without a finite verb to form part of a sentence
by---preposition used with a passive to show the subject does something,makes something happen.

+ing例如:play+ing=playing

No offence:
“There were more than 100,000 accidents happened last year due to the fact that sleep-deprived drivers falling asleep in their car.” ~ grammatically wrong.
1. Replace “happened” with “happening”
2. For the that-cluase, use “fell” instead of “falling”
3 "car " should be plural

2013-11-02 13:04:39 補充：
I beg to differ.

"at the wheel" means "driving the car"
I think it simply adds more weight.

開緊車時睡著了 compare with 在車裡面睡著了

Your sentence is phrased in an incorrect way, and you also missed a verb. I would re-write it as follows:
There were more than 100,000 accidents happened last year due to the fact that sleep-deprived drivers falling asleep in their car. (I don't think "at the wheel" can fully convey the message).

As Raymond as pointed out, the sentence is missing a "verb". I believe it may be "caused", so the sentence may read:

For example, sleep-deprived drivers caused more than 100,000 accidents last year by falling asleep at the wheel.

With the verb "caused", we can now see that "by falling asleep at the wheel" is an "adverbial phrase (of manner)", decribing "how" the "sleep-deprived drivers" caused accidents.

If we use "and they falling asleep", the "and" as a conjunction means adding more information, but does not help explain why "sleep-deprived drivers" cause accidents. Also, if the conjunction "and" is used, it needs to be followed by a "clause" instead of a "phrase", and a "verb" will be required, e.g. "and they would be falling asleep".

Overall, by + present participle (verb + ing) is a prepositional adverbial phrase of manners, to explain "how" the action of the "verb" can happen. It has a similar usage and meaning as "as a result of ".

Examples:
1. How can we relax ?
We can relax by listening to music.
We can relax as a result of listening to music.

2. How can he lose some weight ?
He can lose weight by eating more healthily, and by doing more exercise.
He can lose weight as a result of eating more healthily, and doing more exercises.

3. How can "sleep-deprived drivers" cause accidents?
They can cause accident by falling asleep while driving.
They can cause accidents as a result of falling asleep while driving.



2013-11-01 23:48:16 補充：
Correcton:
1st sentence s/b: "As Raymond has pointed out ...."

2013-11-02 14:43:47 補充：
"at the wheel" = driving a vehicle.

Please double check. I think you have left out the verb of the sentence.