Permutations and combinations are two fundamental concepts in combinatorics, a branch of mathematics concerned with counting, arrangement, and probability. These concepts are crucial in various fields, including statistics, computer science, and physics.

## Permutations (Arrangements)

**Definition:**Ordering of objects.**Key Point:**Order matters.**Example:**'ABC' has permutations like ABC, ACB, BAC, etc.

## Combinations (Selections)

**Definition:**Selection of objects regardless of order.**Key Point:**Order doesn't matter.**Example:**From 'ABC', combinations of 2 letters are AB, AC, BC.

## Permutation Formulas

**1. Permutation of n Distinct Objects**

**Formula:**$P(n) = n!$**Use:**Arrange n different objects.**Example:**4 books arranged in $4!$ ways.

**2. Permutation with Repetition**

**Formula:**$P(n, r) = \frac{n!}{(n-r)!}$**Use:**Choose r from n objects.**Example:**Arrange 3 out of 4 books using $P(4, 3)$ .

## Combination Formulas

1. **Combination of n Distinct Objects**

**Formula:**$C(n, r) = \frac{n!}{r!(n-r)!}$**Use:**Choose r from n objects, order irrelevant.**Example:**3 books from 5 using $C(5, 3)$.

## Applications

**1. Arranging 'MATHS'**

**Steps:**Count letters (5). Use $5!$. Answer: 120 ways.

**2. Forming a 3-member Committee from 7**

**Steps:**Use $C(7, 3) = \frac{7!}{3!(7-3)!}$. Answer: 35 ways.

## Calculation Approach

**Identify Problem Type:**Permutations (order matters) or combinations (order doesn't matter).**Select Formula:**Based on problem type.**Define Variables:**n (total objects), r (objects to arrange/select).**Perform Calculations:**Apply the formula and calculate.**Interpret Results:**Understand and explain outcome.

### Example: Arranging 5 Books

**Problem:**Find arrangements for 5 distinct books.**Formula:**Use permutation: $P(n) = n!$.**Apply:**Here, $n = 5$, calculate $5!$.**Calculate:**$5! = 5 \times 4 \times 3 \times 2 \times 1$.**Result:**120 different arrangements possible.

Written by: Dr Rahil Sachak-Patwa

LinkedIn

Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.