Verbs with two forms following after

[SARDORBEK]

We know that after the verbs begin, start, commence, intend, bother, propose, neglect and a few verbs we can use the infinitive or gerund form of the verb with no or little difference in meaning. I have seen this rule many times in lots of English grammar books. For example, in “Advanced grammar in Use” (M.Hewings) the following rule is given:

The verbs begin, cease, start, and continue can be followed by either a to-infinitive or an an – ing form with little difference in meaning.

Even though it was raining, they continued to play/playing.

I was given a difficult test by some teacher the other day. I had to choose only one answer because of the requirement of the test. The question-test is as follows:
His son had to leave school and started … at an office as a clerk.
a) working
b) to work
c) work
d) worked

In this case, do the answers (a) & (b) contradict each other? Does “little difference in meaning” in the above-mentioned rule apply here? On the whole, what is the significant difference between the infinitive and gerund when using after these verbs?

I always rely on your answers.
Respectfully,
SardorBEK