Comprehensive guide and practice questions for CBSE Class 11 Applied Mathematics curriculum (2026-27).
8.2 Factorial Notation: Properties, Algebraic Manipulation, and Legendre's Theorem
In Applied Mathematics, algorithmic complexity analysis, and combinatorial probability, the factorial is the foundational operator that encapsulates the cumulative product of sequential integers. It enables compact algebraic representation of permutations, combinations, and binomial coefficients.
Fundamental Definitions & Recurrence Properties
Definition of Factorial: For any positive integer \(n in \mathbb{N}\), \(n!\) (read as "\(n\) factorial") is the product of the first \(n\) consecutive natural numbers:
\[ n! = n \times (n - 1) \times (n - 2) \times \dots \times 3 \times 2 \times 1 \]
Fundamental Recurrence Relation:
\[ n! = n \cdot (n - 1)! \quad (\text{for all } n \ge 1) \]
Repeated expansion: \(n! = n(n - 1)(n - 2)! = n(n - 1)(n - 2)(n - 3)!\).
The \(0! = 1\) Identity:
Setting \(n = 1\) into the recurrence relation:
\[ 1! = 1 \cdot (1 - 1)! = 1 \cdot 0! \implies 0! = 1 \]
This standard convention guarantees that combinations and permutations evaluate consistently when selecting 0 items or all items.
Non-Distributivity Warning:
Factorial is strictly non-linear:
\[ (a + b)! \neq a! + b!, \quad (a - b)! \neq a! - b!, \quad (a \cdot b)! \neq a! \cdot b!, \quad \left(\frac{a}{b}\right)! \neq \frac{a!}{b!} \]
Legendre's Formula (Exponent of a Prime in \(n!\)):
The highest power of a prime \(p\) that divides \(n!\) is:
\[ E_p(n!) = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \dots \]
Trailing Zeros: The number of trailing zeros in \(n!\) equals \(E_5(n!)\) because 2s always outnumber 5s.
Key Algebraic Simplification Techniques
Algebraic Expression
Simplification Technique
Result
\(\frac{n!}{(n - 2)!}\)
Expand numerator down to \((n - 2)!\): \(\frac{n(n - 1)(n - 2)!}{(n - 2)!}\)