Permutation & Combination

① Permutation — nPr (Order MATTERS)

Arranging r objects from n distinct objects where order matters.

nPr = n! / (n − r)!

Keywords: arrange, order, rank, sequence, line up, schedule, assign position, 4-digit number

Worked Example

In how many ways can 3 of 5 students be assigned to President, VP, and Secretary?

5P3 = 5! / (5−3)! = 5! / 2! = (5×4×3×2×1)/(2×1) = 60

② Combination — nCr (Order does NOT matter)

Choosing r objects from n distinct objects where order does not matter.

nCr = n! / (r! × (n − r)!)

Keywords: choose, select, committee, group, team, subset, sample, combination, handshake

Worked Example

How many ways can a committee of 3 be chosen from 7 people?

7C3 = 7!/(3!×4!) = (7×6×5)/(3×2×1) = 210/6 = 35

③ Must-Memorize Formulas & Properties

Formula	Value / Rule	Type
0! = 1	By definition	P
nP0 = 1, nPn = n!	Select none or all	P
nC0 = nCn = 1	Choose none or all	C
nCr = nC(n−r)	Symmetry property	C
nCr = nPr ÷ r!	Relationship P & C	C
nCr + nC(r−1) = (n+1)Cr	Pascal's Identity	C
Circular: (n−1)!	n people around a table	P
Repetition allowed: n^r	r slots, n choices each	P
No two adjacent: fix others, fill gaps	Restriction technique	P

④ The Five Core Problem Types

Type	Description	Core Idea
1	Basic Permutation	nPr = n!/(n−r)!
2	Basic Combination	nCr = n!/(r!(n−r)!)
3	Permutation with Restriction	Fix restricted items → arrange rest
4	Combination with Restriction	Total − excluded OR case-by-case
5	Circular / Special	(n−1)! or n^r

Answer Key & Full Solutions