If 1 × 2 × 3 × 4 × ... × n is denoted by n! (n factorial), then simplify the expression 15! - 14! - 13! and express it in factored form.

Introduction / Context:
This question tests the understanding of factorial notation and algebraic manipulation of expressions involving factorials. Factorial identities appear frequently in combinatorics, probability and series expansions, so being comfortable factoring expressions like 15! - 14! - 13! is important for simplifying more complex formulas in aptitude as well as in higher mathematics.

Given Data / Assumptions:

• n! denotes the product 1 × 2 × 3 × ... × n.
• We are given the expression 15! - 14! - 13!.
• The task is to simplify the expression and match it with one of the given factored forms.
• No approximations are involved; it is pure algebraic simplification.

Concept / Approach:
The main idea is to factor out the smallest factorial term that is common to all parts of the expression. Since 13! divides both 14! and 15!, it is natural to factor 13! out of the entire expression. After writing 14! and 15! as multiples of 13!, we can combine the coefficients and factor the result further, if possible. This reduces a large looking expression into a compact product form.

Step-by-Step Solution:
Step 1: Write 15! in terms of 13!: 15! = 15 × 14 × 13!.Step 2: Write 14! in terms of 13!: 14! = 14 × 13!.Step 3: The expression becomes 15! - 14! - 13! = 15 × 14 × 13! - 14 × 13! - 13!.Step 4: Factor out the common 13!: 13! × (15 × 14 - 14 - 1).Step 5: Simplify inside the brackets: 15 × 14 = 210, then 210 - 14 - 1 = 210 - 15 = 195.Step 6: Write 195 as 15 × 13, so the expression becomes 13! × 15 × 13 = 15 × 13 × 13!.
Verification / Alternative check:
As a numerical check, one could compute the approximate values using a calculator or software to verify that 15! - 14! - 13! equals 15 × 13 × 13!. However, the algebraic derivation above is already exact. The factoring step using 13! is standard and reliable, so the expression 15 × 13 × 13! is confirmed as correct.

Why Other Options Are Wrong:
14 × 13 × 13! would correspond to 13! × 182, but we have 13! × 195, so it is too small. 15 × 14 × 14! is much larger and corresponds to 15 × 14 × 14!, which is unrelated. 14 × 12 × 12! is not algebraically equivalent either. Hence only 15 × 13 × 13! matches the simplification.

Common Pitfalls:
One common error is to factor out 15! or 14! incorrectly, leading to complicated or incorrect expressions. Another mistake is forgetting that 14! = 14 × 13! and 15! = 15 × 14 × 13!, which makes the algebra messy. Always factor out the smallest common factorial term to keep calculations neat.

Final Answer:
The simplified expression 15! - 14! - 13! equals 15 × 13 × 13!.