Permutation: Arrangements Where Order Matters
Combination: Selections Where Order Doesn’t Matter
Key Differences Between Permutation and Combination
Practical Problems
- Permutation Problem
- Combination Problem
Mixed Problem Example
Applications
FAQs on Problem on Permutation and Combination

Problem on Permutation and Combination

Q: What is the difference between permutation and combination?

Permutation refers to the arrangement of items where the order matters. For example, arranging 3 books on a shelf is a permutation problem. Combination refers to the selection of items where the order does not matter. For example, choosing 3 books from a set of 5 is a combination problem.

Q: How do I decide whether to use a permutation or a combination formula?

If the problem involves arranging items or if the order of selection is important, use permutation . If the problem involves selecting items and the order of selection is irrelevant, use combination .

Q: How can permutation and combination problems be applied in real life?

Permutation : Planning seating arrangements, arranging passwords, or scheduling tasks. Combination : Selecting a team from a group of candidates, choosing lottery numbers, or forming subsets of items.

Permutation and Combination are important concepts in mathematics that deal with arranging and selecting objects. They’re widely used in areas like probability, computer science, and statistics. Let’s break them down with examples.

Permutation: Arrangements Where Order Matters

When the order of objects matters, we use permutations.

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Formula for Permutation:

P(n, r) = \frac{n!}{(n - r)!}

$P$ $($ $n$ $,$ $r$ $)$ $=$ $($ $n$ $-$ $r$ $)!$ $n$ $!$ $$

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$n$ $n$ : Total number of objects.
$r$ $r$ : Number of objects selected.

Example: If you have 5 books and want to arrange 3 on a shelf:

P(5, 3) = \frac{5!}{(5-3)!} = \frac{120}{2} = 60

$P$ $($ $5$ $,$ $3$ $)$ $=$ $($ $5$ $-$ $3$ $)!$ $5$ $!$ $$ $=$ $2120$ $$ $=$ $60$

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So, there are 60 ways to arrange 3 books out of 5.

Combination: Selections Where Order Doesn’t Matter

When the order doesn’t matter, we use combinations.

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Formula for Combination:

C(n, r) = \frac{n!}{r!(n - r)!}

$C$ $($ $n$ $,$ $r$ $)$ $=$ $r$ $!$ $($ $n$ $-$ $r$ $)!$ $n$ $!$ $$

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$n$ $n$ : Total objects.
$r$ $r$ : Number of objects selected.

Example: If a teacher wants to select 3 students from a group of 7:

C(7, 3) = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35

$C$ $($ $7$ $,$ $3$ $)$ $=$ $3$ $!$ $($ $7$ $-$ $3$ $)!$ $7$ $!$ $$ $=$ $3$ $\times$ $2$ $\times$ $17$ $\times$ $6$ $\times$ $5$ $$ $=$ $35$

So, there are 35 ways to select the students.

Key Differences Between Permutation and Combination

Permutation	Combination
Order matters	Order doesn’t matter
$P(n, r) = \frac{n!}{(n-r)!}$ $P$ $($ $n$ $,$ $r$ $)$ $=$ $($ $n$ $-$ $r$ $)!$ $n$ $!$ $$	$C(n, r) = \frac{n!}{r!(n-r)!}$ $C$ $($ $n$ $,$ $r$ $)$ $=$ $r$ $!$ $($ $n$ $-$ $r$ $)!$ $n$ $!$ $$

Practical Problems

Permutation Problem

How many ways can you arrange the letters in the word “CAT”?

P(3, 3) = \frac{3!}{(3-3)!} = 3! = 6

$P$ $($ $3$ $,$ $3$ $)$ $=$ $($ $3$ $-$ $3$ $)!$ $3$ $!$ $$ $=$ $3$ $!$ $=$ $6$

The arrangements are: CAT, CTA, ACT, ATC, TCA, TAC.

Combination Problem

How many ways can you choose 4 players from a team of 10 if one player is fixed?

C(9, 3) = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84

$C$ $($ $9$ $,$ $3$ $)$ $=$ $3$ $\times$ $2$ $\times$ $19$ $\times$ $8$ $\times$ $7$ $$ $=$ $84$

So, there are 84 ways to form the team.

Mixed Problem Example

How many ways can you select and arrange 3 books from 6?

Select the books:

C(6, 3) = 20

$C$ $($ $6$ $,$ $3$ $)$ $=$ $20$

Arrange the selected books:

P(3, 3) = 6

$P$ $($ $3$ $,$ $3$ $)$ $=$ $6$

Total ways:

20 \times 6 = 120

$20$ $\times$ $6$ $=$ $120$

Applications

Probability: Drawing cards or rolling dice.
Data Organization: Arranging information.
Scheduling: Planning events or tasks.
Cryptography: Generating secure keys.

FAQs on Problem on Permutation and Combination

What is the difference between permutation and combination?

Permutation refers to the arrangement of items where the order matters. For example, arranging 3 books on a shelf is a permutation problem.
Combination refers to the selection of items where the order does not matter. For example, choosing 3 books from a set of 5 is a combination problem.

How do I decide whether to use a permutation or a combination formula?

If the problem involves arranging items or if the order of selection is important, use permutation.
If the problem involves selecting items and the order of selection is irrelevant, use combination.

What are the formulas for permutation and combination?

Permutation formula: $P(n, r) = \frac{n!}{(n - r)!}$ $P$ $($ $n$ $,$ $r$ $)$ $=$ $($ $n$ $-$ $r$ $)!$ $n$ $!$ $$ , where $n$ $n$ is the total number of items and $r$ $r$ is the number of items to arrange.
Combination formula: $C(n, r) = \frac{n!}{r!(n - r)!}$ $C$ $($ $n$ $,$ $r$ $)$ $=$ $r$ $!$ $($ $n$ $-$ $r$ $)!$ $n$ $!$ $$ , where $n$ $n$ is the total number of items and $r$ $r$ is the number of items to select.

How can permutation and combination problems be applied in real life?

Permutation: Planning seating arrangements, arranging passwords, or scheduling tasks.
Combination: Selecting a team from a group of candidates, choosing lottery numbers, or forming subsets of items.