Problem in adjective clauses’ grammar

Solution 1:

Doing Away with One’s Relatives

It appears _(that) your test-giver expects you to produce reduced relative clauses here. This is common, but by no means strictly necessary.

The first sentence supports these possible correct solutions:

All the students Dr. Freeman teaches do well in writing.
All the students that Dr. Freeman teaches do well in writing.
All the students whom Dr. Freeman teaches do well in writing.

The first solution above is the most common. the third the least. It is not that your proposed solution two is wrong, only that is less common than solution one is.

The second sentence supports these possible correct solutions, again ranked from most common to least:

The problem I had expected never occurred.
The problem that I had expected never occurred.
The problem which I had expected never occurred.

Your own proposed solution appears to have a spurious it at the end, which makes no sense here. It is otherwise fine, however.

Relative-pronoun Deletion

What is going on in all these situations is that the relative pronoun _{(that is)} connecting the relative clause to the noun phrase _(that) it applies to is usually omissible in English. Another way of saying this is that the zero relative pronoun is often allowed here.

This is perfectly natural and completely common, but it is never obligatory — despite possible protestations to the contrary from your instructor. I’m sure your instructor knows this, so I am guessing _(that) this was a test to see whether you knew how to delete relatives if you cared to.

Relative pronouns are who, whom, whose, that, and which. Since we are talking about clauses here, there must be both a subject and verb. The relative pronoun connects its relative clause to the noun phrase _(which) the clause modifies.

Object relatives

When the relative pronoun is serving as the object of its clause, as it is here, it can often be safely deleted — and usually is.

The bird that I saw in my courtyard interested me.
The bird which I saw in my courtyard interested me.
The bird I saw in my courtyard interested me.

Or with a phrase verb:

The bird that I was looking for appeared in my courtyard.
The bird which I was looking for appeared in my courtyard.
The bird I was looking for appeared in my courtyard.

Note that phrasal verbs can lead to unnatural pied piping, as it does here:

The bird for which I was looking appeared in my courtyard.

Or using a human/animate subject:

The man that I saw in my courtyard interested me.
The man which I saw in my courtyard interested me.
The man whom I saw in my courtyard interested me. (formally correct, but hard to get right)
The man who I saw in my courtyard interested me. (much more common)
The man I saw in my courtyard interested me.

Those are all perfectly fine, and all are encountered often enough in the wild, so to speak.

Subject relatives

When the relative pronoun is serving as the subject of its clause, it is not in general omissible, as this leads to ungrammatical forms unproduced by native speakers:

The bird that came to my courtyard interested me.
The bird which came to my courtyard interested me.
The bird *came to courtyard interested me. (NOT GRAMMATICAL!)

However, there is one special case that allows for the deletion of the relative pronoun serving as subject: if the verb it governs is a finite inflection of be. In that instance, the relative pronoun and the following be verb are both deleted. This is what linguists refer to as whiz-deletion — which has nothing to do with Gandalf falling into shadow in Moria, even if folks with the whine–wine merger might be momentarily misled. :-)

First, with a prepositional phrase:

The bird that was in my courtyard interested me.
The bird which was in my courtyard interested me.
The bird in my courtyard interested me.

Or with a human/animate agent:

The man that was in my courtyard interested me.
The man who was in my courtyard interested me.
The man in my courtyard interested me.

Second, with a past participle:

The bird that was perched in my courtyard interested me.
The bird which was perched in my courtyard interested me.
The bird perched in my courtyard interested me.

Or with a human/animate agent:

The man that was seated in my courtyard interested me.
The man who was seated in my courtyard interested me.
The man seated in my courtyard interested me.

Third, with a present participle:

The bird that was perching in my courtyard interested me.
The bird which was perching in my courtyard interested me.
The bird perching in my courtyard interested me.

Or with a human agent:

The man that was waiting in my courtyard interested me.
The man who was waiting in my courtyard interested me.
The man waiting in my courtyard interested me.

Whizzers Beware!

Note that whiz deletion can become awkward or even misleading when the relative clause is descriptive instead of restrictive:

My kitten Fluffy, who is always by my pillow when I wake up in the morning, was not there this morning.
My kitten Fluffy, always by my pillow when I wake up in the morning, was not there this morning.

Be careful with whiz deletion in those situations; it may or may not improve the reading.

Another source of trouble from overzealous whiz deletion derives from the participle–gerund duality in English. When there is a present participle involved, whiz deletion can sometimes lead to a misreading. Sometimes this is only on first read, but at other times it is always ambiguous.

This occurs because an -ing verb in English can serve equally well as a participle/adjective as it can a gerund/noun — and if it is a noun, then it can be the head of its own phrase.

It is easy enough to construct a garden path sentence that is misreadable, at least initially:

The genetic risk that is taking its toll here will not be seen for generations.
The genetic risk taking its toll here will not be seen for generations.

In the second version, the reader first construes the subject of the sentence to be risk taking, not risk by itself.

The careful writer avoids whiz deletions that imperil clarity, although the mischievous writer may have other goals.

Possessive relatives

No, these are not greedy family members. :) They are relative pronouns that connect their clauses back to the noun phrase in neither a subject nor an object way, but in a possessive way.

These possessive relatives are usually just whose, but may at times be of which instead. There is no difference between those two, though, because whose is not restricted to animate agents _(in) the way _(that)who and whom are.

One cannot delete possessive relatives at all, because the sentence becomes ungrammatical if one tries.

Compare valid sentences:

The man whose bird had escaped chased it into my courtyard.
A bird whose likes I had never seen appeared in my courtyard.
A bird the likes of which I had never seen appeared in my courtyard.

With ones rendered invalid once their relatives have been done away with:

*The man bird had escaped chased it into my courtyard.
*A bird likes I had never seen appeared in my courtyard.
*A bird the likes I had never seen appeared in my courtyard.

Solution 2:

All the students(i) [ (null/that/~~which~~/who/whom) Dr. Freeman teaches __(i) ] do well in writing.
The problem(i) [ (null/that/which) I had expected __(i) ] never occurred.

The stuff within the brackets is a relative clause. Notice that your relatives have a relativized gap ("__") in them. Those gaps cannot be filled--else their clauses would then no longer be relative clauses.

Your gaps have an antecedent, which in your case are the subjects of the main clause.

Note that "null" means that the relative word could be nothing at all.

And so, for your alternatives, yes, you can insert "that" or "which"; but you cannot insert "them" or "it" because that would fill in the relativized gap.

Solution 3:

In his discussion of relative clauses Swan (Practical English Usage, p 478) states:

As subjects or objects, who(m), which and that replace words like she, him or it: one subject or object in a relative clause is enough:

I've found the car keys. You were looking for them.

I've found the car keys that you were looking for. (NOT ... that you were looking for them.)

And Yule in Explaining English Grammar (p241) comments:

In a number of other languages, there would typically be a pronoun after the verb in these relative clauses. Using such a construction in English results in an ungrammatical sentence:

*Did you enjoy the film which you saw it?

The extra pronoun is called a RESUMPTIVE PRONOUN and may be used by some language learners in the early stages of learning English.

Languages that have resumptive pronouns include Greek and Persian.