Difference Between Permutation and Combination

What is a Permutation?

A permutation is an arrangement of objects in a specific order. If you change the position of even one item, it creates a brand-new permutation.

• Key Phrase: "Arrangement matters."

• Real-life Example: Think of the words "ACT" and "CAT." The words have different meanings because the order is different, even though they employ the same characters. These are two alternative ways to arrange things.

Permutation Formula

To calculate the number of ways to arrange $r$ items out of $n$ total items, we use the permutation formula:

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

• $n$: Total number of items in the set.

• $r$: Number of items being chosen.

• $!$: Factorial (e.g., $4! = 4 \times 3 \times 2 \times 1 = 24$).

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What is a Combination?

A combination is a group of things where the order doesn't matter. You are only picking a group, and the way the group is set up within of it doesn't matter.

• Key Phrase: "Selection matters, order doesn't."

• Real-life Example: If you were making a fruit salad with an apple, a banana, and a cherry. The salad will taste the same no matter which fruit you put in first, the Apple or the Banana. This is just one combination.

Combination Formula

The combination formula is how we figure out how many possibilities there are to choose $r$ things from $n$ total items: 

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

The combination formula has an extra $r!$ that you should notice.$ in the bottom. This "divides out" the numerous ways the same group can be arranged, which means there are fewer possible outcomes than a permutation.

Difference between Permutation and Combination

To truly understand the difference between permutation and combination, let’s look at this comparison table:

Read More - Difference Between Percentage and Percentile, Rules, Examples

Difference Between Permutation and Combination with Examples

Let’s apply these concepts to some scenarios to see how they work in practice.

Example 1: The Race (Permutation)

There are 3 runners: Amit, Biplab, and Chandra. We want to know how many ways they can win Gold and Silver medals.

• Logic: The order matters. Amit getting Gold and Biplab getting Silver is different from Biplab getting Gold and Amit getting Silver.

• Calculation: Here $n=3, r=2$.

• Result: {Amit, Biplab}, {Biplab, Amit}, {Amit, Chandra}, {Chandra, Amit}, {Biplab, Chandra}, {Chandra, Biplab} = 6 Permutations.

Example 2: The Committee (Combination)

From the same 3 people (Amit, Biplab, and Chandra), we want to choose a team of 2 people to go to a meeting.

• Logic: The order does not matter. A team of {Amit, Biplab} is the same as a team of {Biplab, Amit}.

• Calculation: Here $n=3, r=2$.

• Result: {Amit, Biplab}, {Amit, Chandra}, {Biplab, Chandra} = 3 Combinations.

When to Use Permutation and Combination?

The hardest part is figuring out when to utilise permutation and combination when you read a word problem.  Ask yourself this one "Magic Question":

"If I swap the positions of two items in my result, does it create a new scenario?"

• If YES, use Permutation. (Order matters!)

• If NO, use Combination. (Order doesn't matter!)

Use Permutations for:

• Creating passwords or PIN codes.

• Seating arrangements in a row.

• Assigning specific job titles (President, VP, Secretary).

• Arranging letters to make words.

Use Combinations for:

• Picking a hand of cards in a game.

• Selecting a committee or a team.

• Choosing toppings for a pizza.

• Selecting books from a library to take home.

Read More - Difference Between Rational Numbers and Irrational Numbers

Detailed Example for Difference Between Permutation and Combination

Let's look at a 10-person club ($n=10$) to further illustrate the difference between permutation and combination.

Scenario A: You need to pick a President, a Vice President, and a Treasurer.

Since the roles are different, the order in which you pick people matters. (Asha as President is different from Asha as Treasurer).

• Method: Permutation.

• Formula: $P(10, 3) = \frac{10!}{(10-3)!} = 10 \times 9 \times 8 = \mathbf{720\text{ ways}}$.

Scenario B: You need to pick a 3-person cleaning crew.

Since everyone on the crew does the same job, the order doesn't matter. (Asha, Bob, and Cal is the same crew as Cal, Bob, and Asha).

• Method: Combination.

• Formula: $C(10, 3) = \frac{10!}{3!(10-3)!} = \frac{720}{3 \times 2 \times 1} = \mathbf{120\text{ ways}}$.

Common Mistakes to Avoid

Here are the some common mistakes to avoid while working on permutation and combination: 

1. Confusing the English Language: We call it a "combination lock," but in math, it's a "permutation lock." If you type in 3-2-1 instead of 1-2-3, it won't open. The sequence is important!

2. Forgetting to Divide in Combinations: Students often forget the $r!$ in the combination formula, which leads to an answer that is too large.

3. Misidentifying the Set ($n$): Always ensure $n$ is the total number available and $r$ is what you are actually choosing.

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