6.1 Factorial Notation and Properties

Definition of Factorial

For any positive integer $n$, the factorial of $n$ (written $n!$) is defined as:

$n!=n\times (n-1)\times (n-2)\times \cdots \times 3\times 2\times 1$

By convention, $0!=1$.

Examples:

• $5!=5\times 4\times 3\times 2\times 1=120$
• $3!=6$, $1!=1$, $0!=1$

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Key Properties of Factorials

1. Recursive Property

$n!=n\times (n-1)!$

This is the most important property for simplifying factorial expressions.

Example: Simplify $\frac{n!}{(n-2)!}$

$\frac{n!}{(n-2)!}=\frac{n\cdot (n-1)\cdot (n-2)!}{(n-2)!}=n(n-1)$

2. Writing Products as Factorial Fractions

A product of consecutive integers can be written as a ratio of factorials:

$n(n-1)(n-2)\cdots (n-r+1)=\frac{n!}{(n-r)!}$

Example: Write $10\times 9\times 8$ in factorial form.

$10\times 9\times 8=\frac{10!}{7!}$

because dividing $10!$ by $7!$ cancels all factors from $7$ down to $1$.

3. Factorial Equations (Quadratic Form)

When a factorial equation involves $(n+k)!$ and $n!$, expand the larger factorial:

$\frac{(n+2)!}{n!}=(n+2)(n+1)$

So an equation like $(n+2)!=42\cdot n!$ becomes: $(n+2)(n+1)=42⟹n^2+3n+2=42⟹n^2+3n-40=0$

Factoring: $(n+5)(n-8)=0$, so $n=8$ (taking the positive integer solution).

4. Adding Factorial Fractions

To add fractions with factorial denominators, use the largest factorial as the LCD.

Example: $\frac{1}{n!}+\frac{1}{(n+1)!}$

$=\frac{n+1}{(n+1)!}+\frac{1}{(n+1)!}=\frac{n+2}{(n+1)!}$

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Product of Odd Integers

The product of the first $n$ odd integers can be expressed using factorials:

$1\cdot 3\cdot 5\cdots (2n-1)=\frac{(2n)!}{2^n\cdot n!}$

Derivation: Multiply and divide by the product of even integers $2\cdot 4\cdot 6\cdots (2n)$:

$1\cdot 3\cdot 5\cdots (2n-1)=\frac{(2n)!}{2\cdot 4\cdot 6\cdots (2n)}$

Since $2\cdot 4\cdot 6\cdots (2n)=2^n\cdot n!$, we get:

$1\cdot 3\cdot 5\cdots (2n-1)=\frac{(2n)!}{2^n\cdot n!}$

Verification for $n=3$: $1\cdot 3\cdot 5=15$ and $\frac{6!}{2^3\cdot 3!}=\frac{720}{8\cdot 6}=\frac{720}{48}=15$ ✓

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Connection to Permutations

The factorial ratio $\frac{n!}{(n-r)!}$ appears directly in the permutation formula:

$P(n,r)=\frac{n!}{(n-r)!}=n(n-1)(n-2)\cdots (n-r+1)$

This counts the number of ways to arrange $r$ objects chosen from $n$ distinct objects.

Example: The number of ways to arrange 3 books chosen from 7 distinct books is: $P(7,3)=\frac{7!}{4!}=7\times 6\times 5=210$

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