If $a, b, c, \dots, x, y, z$ are $26$ natural numbers, then the value of $(x - a)(x - b)(x - c) \dots (x - y)(x - z)$ is

Concept & Logic This is a classic logical puzzle disguised as an algebra problem. It tests your ability to recognize patterns within a sequence rather than performing actual mathematical calculations. The key is identifying a specific term within the long product chain that forces the entire expression to evaluate to a constant. Step-by-Step Solution * Given expression: $$ (x - a)(x - b)(x - c) \dots (x - y)(x - z) $$ * Step 1: Understand the pattern of the sequence. The first term in each bracket is always the variable $x$. The second term iterates through the alphabet from $a$ to $z$. * Step 2: Trace the sequence mentally. It goes $(x - a)$, $(x - b)$, $(x - c)$, and so on. Since it iterates through the entire alphabet, it must eventually reach the letter $x$ as the second term. * Step 3: Identify the critical bracket. At the 24th position in the sequence, the bracket will be: $$ (x - x) $$ * Step 4: Evaluate the critical bracket. $$ (x - x) = 0 $$ * Step 5: Apply the zero product property. Any number multiplied by $0$ results in $0$. $$ \text{(Product of previous terms)} \times 0 \times \text{(Product of remaining terms)} = 0 $$ Exam Strategy & Shortcut **Look for the Zero Term:** Whenever you see a long sequential product, especially one involving the alphabet or a long string of integers, immediately scan for a term that evaluates to zero. In alphanumeric sequences, look for self-subtraction like $(x - x)$ or $(n - n)$. The moment you realize $(x - x)$ is part of the sequence, the answer is instantly $0$ without any further thought. Common Pitfall The main pitfall is panic. Students see an infinite-looking algebra string with 26 unknown variables and assume it requires complex polynomial expansion or advanced theorems, wasting time on a problem designed to be solved in two seconds via logical deduction. Final Answer **Therefore, the correct answer is 0.**