Mastering Permutation and Combination- Key Formulas and Applications

What Are Permutations and Combinations?

These two concepts trip up more students than almost any other topic in probability and combinatorics. Here's the difference in plain terms:

Permutation = arrangement where order matters. The sequence ABC is different from CBA.

Combination = selection where order does not matter. ABC and CBA are the same selection.

That's it. Everything else in this article builds on that single distinction.

The Core Formulas

Fundamental Counting Principle

Before diving into formulas, understand this: if task A can be done in m ways and task B can be done in n ways, both tasks can be done in m × n ways.

Example: You have 4 shirts and 3 pants. Total outfits = 4 × 3 = 12.

Permutation Formulas

When selecting r items from n items (order matters, no repetition):

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

When repetition is allowed:

P(n,r) = n^r

Combination Formulas

When selecting r items from n items (order does not matter, no repetition):

C(n,r) = n! / [r! × (n - r)!]

This is often written as "n choose r" and shown as (n/r) or nCr.

When repetition is allowed:

C(n+r-1, r) = (n+r-1)! / [r! × (n-1)!]

Special Cases

Permutation vs Combination: When to Use Which

Use this table to quickly identify which concept you need:

Situation Concept Example
Arranging people in a line Permutation How many ways to arrange 5 people?
Selecting a committee Combination How many ways to choose 3 from 10?
Creating passwords or codes Permutation How many 4-digit PINs exist?
Dealing hands of cards Combination How many 5-card poker hands?
Selecting toppings on a pizza Combination Choose 3 from 8 available toppings

How to Solve Problems: Step-by-Step

Step 1: Identify the Problem Type

Ask yourself: Does order matter here?

If you're arranging, ranking, or sequencing → permutation.

If you're choosing, selecting, or grouping → combination.

Step 2: Check for Repetition

Can items be repeated? If yes, use the repetition formulas.

Example without repetition: Selecting lottery numbers (each number can only be chosen once).

Example with repetition: Forming 3-letter words from the alphabet (letters can repeat).

Step 3: Plug Into the Formula

For permutations: P(n,r) = n! / (n-r)!

For combinations: C(n,r) = n! / [r!(n-r)!]

Practical Examples

Example 1: Permutation Without Repetition

Problem: How many ways can 3 people be seated from a group of 10?

Solution:

P(10,3) = 10! / (10-3)! = 10! / 7! = 10 × 9 × 8 = 720

Example 2: Combination Without Repetition

Problem: How many ways to choose 2 toppings from 5 available?

Solution:

C(5,2) = 5! / [2! × 3!] = 120 / 6 = 10

Example 3: Permutation With Repetition

Problem: How many 4-letter strings can be formed from A, B, C with repetition allowed?

Solution:

P(3,4) with repetition = 3^4 = 81

Real-World Applications

Common Mistakes to Avoid

Quick Reference Cheat Sheet

Formula Type Formula When to Use
Permutation (no repeat) n!/(n-r)! Order matters, no repeats
Permutation (with repeat) n^r Order matters, with repeats
Combination (no repeat) n!/[r!(n-r)!] Order doesn't matter, no repeats
Combination (with repeat) (n+r-1)!/[r!(n-1)!] Order doesn't matter, with repeats

How to Remember the Formulas

The combination formula is literally the permutation formula divided by r!. That's because combinations ignore order, and there are r! ways to arrange any r items.

So:

If you forget the combination formula, just derive it from permutations. 🎯