If $1 \times 2 \times 3 \times ........ \times n$ is denoted by $\lfloor n$, then $\lfloor 8 - \lfloor 7 - \lfloor 6$ is equal to

### Concept & Formula This question tests the fundamental property of factorials. The symbol $\lfloor n$ is an older notation representing $n!$ (n factorial). The core formula used to simplify factorials is: $$n! = n \times (n-1)!$$ ### Step-by-Step Solution * **Given:** * The expression: $\lfloor 8 - \lfloor 7 - \lfloor 6$ (which translates to $8! - 7! - 6!$) * **Calculation / Deduction:** 1. To subtract these terms easily, express the larger factorials in terms of the smallest factorial in the expression ($6!$). $$8! = 8 \times 7 \times 6! = 56 \times 6!$$ $$7! = 7 \times 6!$$ 2. Substitute these expanded forms back into the original expression: $$(56 \times 6!) - (7 \times 6!) - (1 \times 6!)$$ 3. Factor out the common term, $6!$: $$6! \times (56 - 7 - 1)$$ 4. Simplify the numbers inside the bracket: $$56 - 7 - 1 = 48$$ So, the expression becomes $48 \times 6!$. 5. Look at the options. We need to factor $48$ into $6 \times 8$ to match the format of the given answers: $$48 \times 6! = 6 \times 8 \times \lfloor 6$$ ### Exam Strategy & Shortcut Whenever you see addition or subtraction of consecutive factorials, immediately factor out the smallest one. $8! - 7! - 6! \rightarrow 6!(8 \times 7 - 7 - 1) = 6!(56 - 8) = 6! \times 48$. Scanning the options, only option (c) has $6 \times 8 = 48$ multiplied by $\lfloor 6$. This takes under 20 seconds. ### Common Pitfall Students unfamiliar with the old factorial notation $\lfloor n$ might mistake it for an absolute value, floor function, or right-angle bracket, leading to nonsensical arithmetic like $8 - 7 - 6$. Always recognize this symbol in Indian competitive exams as a factorial. ### Final Answer Therefore, the correct answer is $6 \times 8 \times \lfloor 6$.