To simplify the expression `16 - [11 - {8 - : (17 + 3 x 2 - 19)}]`, we will follow the BODMAS rule, which stands for Brackets, Orders (i.e., powers and roots, etc.), Division, Multiplication, Addition, and Subtraction.
Let's break it down step by step:
### Step 1: Solve the innermost brackets
We start with the expression inside the parentheses:
`(17 + 3 x 2 - 19)`
1. **Calculate the multiplication first:**
- \(3 \times 2 = 6\)
Now the expression becomes:
\[
(17 + 6 - 19)
\]
### Step 2: Continue simplifying the parentheses
2. **Now, perform the addition and subtraction from left to right:**
- \(17 + 6 = 23\)
- \(23 - 19 = 4\)
So, we have:
\[
(17 + 3 \times 2 - 19) = 4
\]
### Step 3: Substitute back into the expression
Now substitute this result back into the original expression:
\[
16 - [11 - {8 - : 4}]
\]
### Step 4: Solve the curly brackets
Now we need to simplify the curly brackets:
\[
{8 - : 4}
\]
3. **Perform the division:**
- \(8 \div 4 = 2\)
So, the curly brackets become:
\[
{8 - 2} = 6
\]
### Step 5: Substitute back into the expression
Now substitute this result back into the expression:
\[
16 - [11 - 6]
\]
### Step 6: Solve the square brackets
Now simplify the square brackets:
\[
[11 - 6]
\]
4. **Perform the subtraction:**
- \(11 - 6 = 5\)
So, we have:
\[
[11 - 6] = 5
\]
### Step 7: Final substitution
Now substitute this result back into the expression:
\[
16 - 5
\]
### Step 8: Final calculation
5. **Perform the final subtraction:**
- \(16 - 5 = 11\)
Thus, the simplified result of the expression is:
\[
\boxed{11}
\]
