Permutations and Combinations

Both Permutations $nPr$ and Combinations $nCr$ are counting techniques used to determine how many ways selections or arrangements can be made from a set of objects.

Concept	Use when
Permutation $nPr$	Order matters
Combination $nCr$	Order does NOT matter

Permutations — ( $nPr$ )

Permutation is an arrangement of objects where order is important.

Example: Arranging students in a line, ranking winners, passwords.

Formula

$\boxed{nPr = \frac{n!}{(n-r)!}}$

where

(n) = total number of objects
(r) = number of objects selected
(!) denotes factorial

Example 1: Simple Permutation

Question:
How many ways can 3 students be arranged from 5 students?

Solution:
$5P3 = \frac{5!}{(5-3)!} = \frac{5!}{2!} = 5 \times 4 \times 3 = 60$

✔ Order matters, so permutation is used.

Example 2: Letter Arrangement

Question:
How many 2-letter words can be formed from A, B, C?

Solution:
$3P2 = \frac{3!}{1!} = 6$

Possible arrangements:
AB, BA, AC, CA, BC, CB

Combinations — ( $nCr$ )

Combination is a selection of objects where order is not important.

Example: Selecting committee members, choosing lottery numbers.

Formula

$\boxed{nCr = \frac{n!}{r!(n-r)!}}$

Example 3: Simple Combination

Question:
How many ways can 3 students be chosen from 5 students to form a team?

Solution:
$5C3 = \frac{5!}{3!2!} = 10$

✔ Order does not matter, so combination is used.

Example 4: Comparing with Permutation

From A, B, C:

Permutations of 2 letters = 6
Combinations of 2 letters = 3

Combinations:
AB, AC, BC

Order like AB and BA are considered same.

Relationship Between $nPr$ and $nCr$

$\boxed{nPr = nCr \times r!}$

This shows:

A permutation is a combination arranged in all possible orders.

Practical Comparison Example

Question:
A class has 10 students.

(a) Choose a captain and vice-captain

Order matters → Permutation

$10P2 = 10 \times 9 = 90$

(b) Choose 2 class representatives

Order does not matter → Combination

$10C2 = \frac{10 \times 9}{2} = 45$

Key Properties of $nCr$

Symmetry property
$\boxed{nCr = nC(n-r)}$

Example:
$5C2 = 5C3 = 10$

Edge values
$nC0 = nCn = 1$

Summary

Feature	$nPr$	$nCr$
Order matters	Yes	No
Used for	Arrangements	Selections
Formula	$\frac{n!}{(n-r)!}$	$\frac{n!}{r!(n-r)!}$
Example	Ranking, seating	Teams, committees

Permutations — ()

Formula

Example 1: Simple Permutation

Example 2: Letter Arrangement

Combinations — ()

Formula

Example 3: Simple Combination

Example 4: Comparing with Permutation

Relationship Between and

Practical Comparison Example

(a) Choose a captain and vice-captain

(b) Choose 2 class representatives

Key Properties of

Summary

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Permutations — ( $nPr$ )

Combinations — ( $nCr$ )

Relationship Between $nPr$ and $nCr$

Key Properties of $nCr$