Two of the quantifiers above can be used
without of:

all of the answers / all the answers

both of their children / both their children

Some of the other quantifiers can be used
without of if the noun after the quantifier is
not preceded by an article, pronoun, or
other determiner:

any of the answers /
any answers

a few of his friends /
a few friends

few of his friends /
few friends

a little of the soup /
a little soup

little of the soup /
little soup

many of her classmates /
many classmates

most of her classmates /
most classmates

some of my books /
some books

When the nouns after these quantifiers do not
have an article, pronoun, or determiner, they
are being referred to in a general way. When
the nouns are preceded by an article, pronoun,
or determiner, the reference is specific:

any answers = any answers in general

any of the answers = any of a specific
set or group of answers

many classmates = "classmates" is used
in a general way

many of her classmates ="classmates"
is used in a specific way: only her classmates
are in the group being referred to

Some of the quantifiers above must be used
with an article, pronoun, or other determiner
preceding the noun which follows of:

right:
neither of the students

wrong:
*neither of students, *neither students

right:
none of the reasons

wrong:
*none of reasons

right:
none of the money

wrong:
*none of money

right:
part of a group

wrong:
*part of group

right:
much of the time

wrong:
*much of time

If the quantifiers are changed, some of them
can be used with singular countable nouns:

neither student

no reason

no money

much time

The meaning may, however, be different.
(Compare "none of the reasons" with "no
reason"; compare "none of the money" with
"no money"; compare "much of the time
with "much time.")

The quantifiers "a great deal of" and "a lot of"
may be used in both a general and a specific
way, but of is necessary in both cases:

a great deal of money /
a great deal of my money

a lot of money /
a lot of the money

a lot of people /
a lot of the people