Understanding Permutations and Combinations Made Simple

Permutation and combination are systematic methods in mathematics to count the ways of arranging or selecting objects in a set, with or without attaching importance to the order of selection or arrangement.

Permutation: Definition, Notation, and Standard Formulae

Definition: A permutation is an arrangement of $r$ objects chosen from $n$ distinct objects, where the order of arrangement is significant.

The number of possible permutations is denoted as $P(n, r)$ or $_nP_r$, and is given by the formula \[ {}_nP_r = \frac{n!}{(n-r)!} \] where $0 \leq r \leq n$ and $n!$ denotes the factorial of $n$.

If all $n$ objects are arranged, the total number of possible arrangements is $n!$. When some objects are identical, the arrangement count is reduced appropriately.

For permutations with repetition allowed, the number of possible arrangements of $r$ objects from $n$ types is $n^r$.

Combination: Meaning and Counting Principle

Definition: A combination is a selection of $r$ objects from $n$ distinct objects, where the order of selection is disregarded.

The number of combinations, denoted as $C(n,r)$ or ${}_nC_r$, is given by \[ {}_nC_r = \frac{n!}{r!(n-r)!} \] with $0 \leq r \leq n$.

The fundamental relation between permutation and combination is $_nP_r = {}_nC_r \cdot r!$.

When repetition is allowed in selection, the number of combinations of $r$ objects from $n$ types is $\displaystyle {n + r - 1 \choose r}$.

Algebraic Properties and Standard Results

Result: ${}_nC_r = {}_nC_{n-r}$ for all valid $n,r$. This follows from the definition and the symmetry of combinations.

Another identity commonly used in problems is: \[ {}_nC_r + {}_nC_{r-1} = {}_{n+1}C_r \] which is the basis for Pascal’s triangle.

For permutations, a basic recurrence is \[ {}_nP_r = n \cdot {}_{n-1}P_{r-1} \] which builds arrangement counts inductively.

For non-distinct objects, if $n$ objects contain $p$ indistinguishable of one kind, $q$ of another, ..., the number of distinct permutations is $\displaystyle \frac{n!}{p!\,q!\,\ldots}$.

Distinguishing Ordering: Permutation versus Combination

A typical exam confusion occurs when students neglect whether sequence is relevant; applications demand strict adherence to distinction between arrangement and selection.

For reference on exam-focused patterns, see Permutations And Combinations Important Questions.

Evaluation of Standard Problems in Permutations and Combinations

Example: Calculate the number of ways to arrange 5 persons on 3 chairs.

Given: $n = 5$, $r = 3$.

By formula, $_5P_3 = \dfrac{5!}{(5-3)!} = \dfrac{120}{2!} = 60$ ways.

Example: Find the number of ways to select 2 items from 5 distinct items.

Given: $n = 5$, $r = 2$.

By formula, $_5C_2 = \dfrac{5!}{2!3!} = \dfrac{120}{12} = 10$ ways.

For more problem types, visit Permutation And Combination Problems.

Example: Determine $n$ for which $_nP_5 = 42 \cdot {}_nP_3$, $n > 4$.

By expansion: $n(n-1)(n-2)(n-3)(n-4) = 42n(n-1)(n-2)$, dividing both sides by $n(n-1)(n-2)$ gives $(n-3)(n-4) = 42$; solving $n^2 - 7n + 12 = 42$ yields $n^2 - 7n - 30 = 0$, so $n = 10$ (rejection of negative root).

Example: A committee of 5 is selected from 8 women and 6 men with at least 3 women. Compute the number of possible committees.

Case 1: 3 women, 2 men: $_8C_3 \cdot {}_6C_2 = 56 \cdot 15 = 840$.

Case 2: 4 women, 1 man: $_8C_4 \cdot {}_6C_1 = 70 \cdot 6 = 420$.

Case 3: 5 women, 0 men: $_8C_5 = 56$.

Total number of committees $= 840 + 420 + 56 = 1316$.

For a detailed summary of theory and examples, see Permutations And Combinations Revision Notes.

Counting Problems in Typical Examinations

Permutations and combinations are frequently applied in counting the number of ways events can occur and are integral to probability analysis and discrete mathematics examinations.

• Number of arrangements with constraints
• Selection under specified conditions
• Arrangements involving repeated elements
• Counting divisibility arrangements
• Application to committee formation
• Counting arrangements of digits/letters
• Seating or lineup problems
• Inclusion of identical objects

Related problems and worksheet resources are available at Permutation And Combination.

For graphical and advanced polynomial application, see also Graph Of Quadratic Polynomial.