Permutations and Combinations Explained: Formulas and Examples

Permutations and combinations are fundamental counting techniques in mathematics. They help you determine the number of ways to arrange or select items from a set. Understanding the difference between the two is crucial for probability, statistics, and many real-world applications from lottery odds to password security.

This guide explains both concepts in depth, provides formulas with worked examples, and clarifies when to use each one.

The Fundamental Counting Principle

Before diving into permutations and combinations, you need the fundamental counting principle: if one event can occur in ways and a second independent event can occur in ways, then the two events together can occur in ways.

Example 1: Counting Principle

A restaurant offers 3 starters and 5 main courses. How many different two-course meals are possible?

Factorials

Both permutation and combination formulas use factorials. The factorial of a non-negative integer , written , is the product of all positive integers up to :

By convention, .

What Are Permutations?

A permutation is an arrangement of items where order matters. The word “ABC” is different from “CBA” because the order of the letters has changed.

Permutations of All Items

The number of ways to arrange all items is:

Example 2: Arranging Books

How many ways can you arrange 6 books on a shelf?

Permutations of r Items from n

The number of ways to choose and arrange items from a set of items is:

Example 3: Selecting a Committee with Roles

From 10 people, how many ways can you choose a president, vice president, and secretary?

Permutations with Repetition

If you can reuse items (like digits in a PIN), the number of arrangements of items chosen from with repetition is:

Example 4: PIN Codes

How many 4-digit PIN codes are possible using digits 0-9?

Permutations of Identical Items

When some items are identical, you divide by the factorials of the counts of identical items to avoid overcounting:

Example 5: Letters in MISSISSIPPI

How many distinct arrangements of the letters in MISSISSIPPI?

There are 11 letters: M(1), I(4), S(4), P(2).

What Are Combinations?

A combination is a selection of items where order does not matter. Choosing players A, B, and C for a team is the same as choosing C, A, and B.

The number of ways to choose items from items (without regard to order) is:

Example 6: Choosing a Team

From 12 players, how many ways can you select a team of 5?

Combinations with Repetition

When repetition is allowed (e.g., choosing scoops of ice cream where you can repeat flavours), the formula becomes:

Example 7: Ice Cream Scoops

An ice cream shop has 5 flavours. How many ways can you choose 3 scoops if you can repeat flavours?

Permutations vs Combinations: How to Tell

The key question is: does order matter?

• Order matters (arrangements, rankings, passwords, sequences): use permutations.
• Order does not matter (teams, groups, selections, committees without roles): use combinations.

The relationship between the two formulas shows this clearly:

Each combination of items can be arranged in ways, which is why the number of permutations is always larger than (or equal to) the number of combinations.

Pascal's Triangle

Pascal's triangle is a triangular array where each entry is the sum of the two entries directly above it. The entries are the binomial coefficients :

This recursive property is known as Pascal's rule and is useful for computing combinations without factorials.

Applications in Probability

Example 8: Lottery Odds

A lottery requires you to pick 6 numbers from 49. What are the odds of winning?

The probability of winning is , or about 1 in 14 million.

Example 9: Card Hands

How many 5-card poker hands can be dealt from a standard 52-card deck?

The Binomial Theorem

Combinations appear in the binomial theorem, which expands powers of a binomial expression:

Example 10: Expand

Common Mistakes

1. Using permutations when combinations are needed. If the problem says “choose”, “select”, or “pick” without mentioning order or arrangement, it is likely a combination problem.
2. Forgetting the repetition case. Standard formulas assume items cannot be reused. Check whether repetition is allowed.
3. Overcounting identical items. When items are indistinguishable, divide by the appropriate factorial.
4. Misapplying the counting principle. Ensure events are truly independent before multiplying.

Try It Yourself

Calculate permutations and combinations instantly with our free Permutation & Combination Calculator. It handles all variations including repetition, with step-by-step solutions and formula breakdowns. For related probability problems, try our Probability Calculator to compute event likelihoods.