Simplify: $1 - [1 - \{1 - (1 - \overline{1 - 1})\}]$

Concept & Rule This problem strictly tests the "V" in V-BODMAS (Vinculum, Brackets, Order, Division, Multiplication, Addition, Subtraction). A vinculum is a horizontal bar placed over an expression to indicate that it should be grouped and evaluated before any other brackets. Step-by-Step Solution * Given expression: $$ 1 - [1 - \{1 - (1 - \overline{1 - 1})\}] $$ * Step 1: Evaluate the Vinculum (the bar over $1 - 1$) first. $$ \overline{1 - 1} = 0 $$ Substitute this back into the expression: $$ 1 - [1 - \{1 - (1 - 0)\}] $$ * Step 2: Evaluate the innermost parentheses `( )`. $$ (1 - 0) = 1 $$ Substitute this back: $$ 1 - [1 - \{1 - 1\}] $$ * Step 3: Evaluate the curly braces `{ }`. $$ \{1 - 1\} = 0 $$ Substitute this back: $$ 1 - [1 - 0] $$ * Step 4: Evaluate the square brackets `[ ]`. $$ [1 - 0] = 1 $$ Substitute this back into the final expression: $$ 1 - 1 $$ * Step 5: Perform the final subtraction. $$ 1 - 1 = 0 $$ Exam Strategy & Shortcut **Inward-Out Tracking:** With nested brackets involving only $1$s and subtraction, you can track the parity (alternating $1$ and $0$). Vinculum: $0$. Next out: $1 - 0 = 1$. Next out: $1 - 1 = 0$. Next out: $1 - 0 = 1$. Final step: $1 - 1 = 0$. By keeping a running mental tally, you can evaluate the entire nest in seconds without writing out every step. Common Pitfall The most frequent error is misinterpreting the vinculum as a negative sign or ignoring it entirely. If the vinculum is ignored, the innermost bracket becomes $(1 - 1 - 1) = -1$, leading to a completely different chain of calculations: $\{1 - (-1)\} = 2$, $[1 - 2] = -1$, and finally $1 - (-1) = 2$, leading students straight to incorrect option (c). Final Answer **Therefore, the correct answer is 0.**