MathematicsChapter 6: Permutation and Combination
FACTORIAL n!
n! (n factorial, Kramp's notation in the book) is 1*2*3*...*n for a positive integer n.
Learning objectives
- 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.
Definition
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.
Where / symbols
- FACTORIAL n!
- n! (n factorial, Kramp's notation in the book) is 1*2*3*...*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.
Relationship
- n! = 12...*n
- 0! = 1
Overview
Compute, then cancel:
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.
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.
Examples 8!/6! = 8*7 · (n+2)!/n! = (n+1)(n+2) · Leave 0! as 1
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
How to answer a factorial n! question
- 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.
Worked example
- 10! / 8!.
- 10! = 10*9*8!
- so 10!/8! = 10*9 = 90
- do not compute 10! in full
In simple words
0! = 1. If you put 0, nCr formulas break at r=0.
Key points
- n! = 12...*n
- 0! = 1
- build from smaller
- write extra factors
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.FACTORIAL n!
Sticky Notes · Quick Revision
Core Idea
- n!
- (n factorial, Kramp's notation in the book) is 123...n for a positive integer n.
- 0!
1. VALUES
- 1!=1, 2!=2, 3!=6, 4!=24, 5!=120, 6!=720.
- n!
- = n * (n-1)!.
- A calculator n!
2. CANCEL IN RATIOS
- (n+1)!
- / n!
- = n+1.
- n!
Worked Example
- Worked: 10! / 8!.
- 10! = 10*9*8!
- so 10!/8! = 10*9 = 90
- do not compute 10! in full
Common Misconception
- 0! = 1. If you put 0, nCr formulas break at r=0.
FBISE Exam Check
- State the rule for factorial n! before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- 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.
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