$\text{Find the value of }[18-3\times \{3-3\times (3÷\frac{1}{3}-3)+3÷\frac{1}{3}+3\}-4\times 2].$

The correct answer is 19.

To evaluate the given mathematical expression, we follow the BODMAS / PEMDAS rule (Order of Operations):

1. Brackets / Parentheses (first innermost parentheses ( ), then curly braces { }, and lastly square brackets [ ])
2. Orders / Exponents
3. Division and Multiplication (from left to right)
4. Addition and Subtraction (from left to right)

The expression to simplify is:

$[18-3\times \{3-3\times (3÷\frac{1}{3}-3)+3÷\frac{1}{3}+3\}-4\times 2]$

Step 1: Evaluate the fraction divisions inside the expression.

We know that dividing by a fraction is equivalent to multiplying by its reciprocal:

$3÷\frac{1}{3}=3\times 3=9$

Step 2: Simplify inside the inner round brackets ( ).

Substitute $3÷\frac{1}{3}=9$ into $(3÷\frac{1}{3}-3)$:

$(9-3)=6$

Step 3: Simplify inside the curly braces { }.

Replace the inner parentheses term with 6 and the other division term with 9 in the curly braces:

$\{3-3\times 6+9+3\}$

Perform multiplication first:

$3\times 6=18$

So the expression inside the curly braces becomes:

$\{3-18+9+3\}$

Evaluating left to right:

$3-18=-15$

$-15+9=-6$

$-6+3=-3$

Step 4: Substitute back into the square brackets [ ].

Now, replace the curly braces portion with -3:

$[18-3\times (-3)-4\times 2]$

Step 5: Perform multiplications inside the square brackets.

$3\times (-3)=-9$

$4\times 2=8$

Substituting these products back into the brackets:

$[18-(-9)-8]$

Step 6: Simplify the final additions and subtractions.

Since subtracting a negative number is equal to adding its positive counterpart:

$18-(-9)=18+9=27$

Finally, subtract 8:

$27-8=19$

Thus, the final value of the expression is 19.