Why E is wrong : "Thrice is an adverb and can not modify a noun. "

D is correct because double/triple are adjectives, so can modify noun.

For further clarity :

"Twice": "Twice" CANNOT function as an object of the preposition such as "by".
"Twice" is an adverb.
Example:
"... increased by more than twice .." -- WRONG!

On a similar note, "DOUBLE" is an adjective and can not modify verb, but since "Twice" is an adverb it can modify Verb.
Example:-
The same amount of acreage produces double the apple.... -- WRONG
The same amount of acreage produces twice as many apples..... -CORRECT

"twice as many as":

"..., twice as many as ..." is an APPOSITIVE modifier. Appositives must modify some noun that comes immediately before the comma.
It helps ?

To continue on this :

Examples:
"... rose sharply in 1990, twice as many as ... " -- WRONG!!
"... rose sharply in 1990, doubling the increase of the previous year." -- RIGHT!!

Long winding is not the one which decides here .

A
three times as many as 1852

three times as many as WHAT ??- Not clear hence in correct

B
three times as much as 1852

three times as MUCH as - we are talking about countables here, hence usage of much incoorect

C
triple what it was in 1852

IT refers to the noun preceding it here tripple hence in correct

D
triple the figure for 1852

Clearly mentions that the population of lawyers have increased by three times

E
thrice the number that was recorded in 1852

Wordier, Passive voice, hence incorrect


Hope this helps !