FACTORIAL n!

FACTORIAL n!

What is it?

n! (n factorial, Kramp's notation in the book) is 123...n for a positive integer n. 0! is defined as 1 (empty product, school). 5! = 120. Factorials grow fast. Meaning: n! counts how many ways to line up n different objects. You will cancel factorials in nPr and nCr by writing them as products and cancelling matching pieces.

In simple words

• n!
• (n factorial, Kramp's notation in the book) is 123...n for a positive integer n.
• 0!
• 0! = 1. If you put 0, nCr formulas break at r=0.

Important terms

FACTORIAL n!: n! (n factorial, Kramp's notation in the book) is 123...n for a positive integer n. 0! is defined as 1 (empty product, school). 5! = 120. Factorials grow fast. Meaning: n! counts how many ways to line up n different objects. You will cancel factorials in nPr and nCr by writing them as products and cancelling matching pieces.

1. VALUES: 1!=1, 2!=2, 3!=6, 4!=24, 5!=120, 6!=720. n! = n * (n-1)!. A calculator n! button is fine in a paper that allows it; still show a cancel line in algebra.

2. CANCEL IN RATIOS: (n+1)! / n! = n+1. n! / (n-2)! = n(n-1) if n>=2. Never cancel inside n! as if it were n * !. Write the extra factors out.

What you should learn

• State the main ideas and notation used in factorial n!.
• Apply the rules of factorial n! in a clear, step-by-step solution.
• Check signs, conditions, and the final answer for mathematical accuracy.

Examples

• 6! = 6*5!
• 10! huge
• 0! = 1 not 0

VALUES

1!=1, 2!=2, 3!=6, 4!=24, 5!=120, 6!=720. n! = n * (n-1)!. A calculator n! button is fine in a paper that allows it; still show a cancel line in algebra.

CANCEL IN RATIOS

(n+1)! / n! = n+1. n! / (n-2)! = n(n-1) if n>=2. Never cancel inside n! as if it were n * !. Write the extra factors out.

Full school-level explanation

n! (n factorial, Kramp's notation in the book) is 123...n for a positive integer n. 0! is defined as 1 (empty product, school). 5! = 120. Factorials grow fast. Meaning: n! counts how many ways to line up n different objects. You will cancel factorials in nPr and nCr by writing them as products and cancelling matching pieces.

Compute, then cancel:

WHAT — FACTORIAL n!

n! (n factorial, Kramp's notation in the book) is 123...n for a positive integer n. 0! is defined as 1 (empty product, school). 5! = 120. Factorials grow fast. Meaning: n! counts how many ways to line up n different objects. You will cancel factorials in nPr and nCr by writing them as products and cancelling matching pieces.

RULE — 1. VALUES

1!=1, 2!=2, 3!=6, 4!=24, 5!=120, 6!=720. n! = n * (n-1)!. A calculator n! button is fine in a paper that allows it; still show a cancel line in algebra.

METHOD — 2. CANCEL IN RATIOS

(n+1)! / n! = n+1. n! / (n-2)! = n(n-1) if n>=2. Never cancel inside n! as if it were n * !. Write the extra factors out.

WORKED EXAMPLE

Worked: 10! / 8!. 10! = 1098! so 10!/8! = 10*9 = 90 do not compute 10! in full

VALUES

1!=1, 2!=2, 3!=6, 4!=24, 5!=120, 6!=720. n! = n * (n-1)!. A calculator n! button is fine in a paper that allows it; still show a cancel line in algebra.

Remember

• build from smaller

Examples

• 6! = 6*5!
• 10! huge
• 0! = 1 not 0

CANCEL IN RATIOS

(n+1)! / n! = n+1. n! / (n-2)! = n(n-1) if n>=2. Never cancel inside n! as if it were n * !. Write the extra factors out.

Remember

• write extra factors

Examples

• 8!/6! = 8*7
• (n+2)!/n! = (n+1)(n+2)
• Leave 0! as 1

How to answer a factorial n! question

How it works

• Identify the given quantities, notation, and required result.
• Choose the matching rule and write it before substitution.
• Work one algebraic step at a time, keeping brackets and signs clear.
• Check restrictions, simplify the result, and verify it where possible.

Remember

• n! = 12...*n
• 0! = 1
• 90
• 0! is 1
• build from smaller
• write extra factors
• n! = 12...*n
• 0! = 1

How the ideas connect

Worked: 10! / 8!. 10! = 1098! so 10!/8! = 10*9 = 90 do not compute 10! in full

Deeper understanding

Go beyond memorising FACTORIAL n!. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked: 10! / 8!. 10! = 1098! so 10!/8! = 10*9 = 90 do not compute 10! in full

Common mistakes

Exam focus

Core Idea

1. VALUES

2. CANCEL IN RATIOS

Worked Example

Exam focus

Quick revision

• n! = 12...*n
• 0! = 1
• 90
• 0! is 1
• FACTORIAL n!: n! (n factorial, Kramp's notation in the book) is 123...n for a positive integer n. 0! is define
• 1. VALUES: 1!=1, 2!=2, 3!=6, 4!=24, 5!=120, 6!=720. n! = n * (n-1)!. A calculator n! button is fine in a paper

0! = 1. If you put 0, nCr formulas break at r=0.

For long answers on FACTORIAL n!: define clearly → explain the main parts → give one example → state one exam caution.