Mathematics IUnit 37 min read

Permutations & Combinations – counting principles, formulas, and applications

Unit 3 of Mathematics I explains the fundamental counting techniques, derives permutation and combination formulas, works through typical problems, and shows how these ideas power everyday services like eSewa, Daraz, and Google.

Padhai · Based on the official syllabus and past papers
Updated 1 Oct 2026

Key points

Permutations count ordered arrangements; combinations count unordered selections.
$nP_r = \dfrac{n!}{(n-r)!}$ and $nC_r = \dfrac{n!}{r!(n-r)!}$ are the core formulas.
Circular permutations divide linear counts by the number of positions, while repetitions modify the factorial denominator.
The relation $nP_r = nC_r \times r!$ links the two concepts.
Real‑world systems such as transaction IDs, delivery queues, and search result rankings are built on these counting principles.

1. Core definitions

Concept	What it counts	Formula (no repetition)
Permutation	Ordered arrangement of $r$ objects chosen from $n$ distinct objects	$^{n} P_{r} = \frac{n !}{( n - r )!}$
Combination	Unordered selection of $r$ objects from $n$ distinct objects	$^{n} C_{r} = \frac{n !}{r ! ( n - r )!}$

Factorial $n!$ = product of all positive integers ≤  $n$ ; by convention $0! = 1$ .

2. Permutations

2.1 Linear permutations (no repetition)

When the order matters and each object can be used at most once, the first position has $n$ choices, the second $n - 1$ , …, the $r^{t h}$ position $n - r + 1$ . Multiplying gives the formula above.

Worked example 1 – arranging books

Five different books ${B_{1}, B_{2}, B_{3}, B_{4}, B_{5}}$ are to be placed on a shelf.

Number of possible orders = $5 P_{5} = 5! = 120$ .

2.2 Permutations with repetition

If each of the $n$ objects may be reused, each position always has $n$ choices:

$Number of arrangements = n^{r} .$

Example: A 4‑digit PIN using digits 0‑9 (repetition allowed) → $1 0^{4} = 10, 000$ possibilities.

2.3 Circular permutations

When objects are placed around a circle, rotations are considered identical. For $n$ distinct objects:

$Circular permutations = \frac{( n - 1 )!}{( n - r )!} (r = n) .$

Example: Seating 6 friends around a round table → $(6 - 1)! = 120$ distinct seatings.

2.4 Permutations of a multiset

If some objects repeat, divide by the factorial of each repeat count:

$\frac{n !}{n _{1} ! n _{2} ! \dots n _{k} !} .$

Example: Arrange the letters of “BANANA”.
$n = 6, n_{B} = 1, n_{A} = 3, n_{N} = 2$

$\frac{6 !}{1 ! 3 ! 2 !} = \frac{720}{12} = 60 distinct arrangements .$

3. Combinations

3.1 Linear combinations (no repetition)

When order does not matter, we first count permutations and then divide by the number of ways to order the chosen $r$ objects ( $r!$ ):

$^{n} C_{r} = \frac{^{n} P _{r}}{r !} = \frac{n !}{r ! ( n - r )!} .$

Worked example 2 – committee selection

From 8 volunteers, choose a 3‑member committee.

$^{8} C_{3} = \frac{8 !}{3 ! 5 !} = \frac{40320}{6 \cdot 120} = 56 possible committees .$

3.2 Combinations with repetition

If an object may be chosen more than once (e.g., selecting 4 fruits from an unlimited supply of 3 types), we use “stars and bars”:

$^{n} C_{r}^{(rep)} = (r n + r - 1) = \frac{( n + r - 1 )!}{r ! ( n - 1 )!} .$

Example: Choose 4 scoops of ice‑cream from 3 flavours → $(4 3 + 4 - 1) = (4 6) = 15$ ways.

3.3 Pascal’s Triangle

Each entry $(r n)$ equals the sum of the two entries above it:

$(r n) = (r - 1 n - 1) + (r n - 1) .$

The triangle provides quick lookup for small $n$ and illustrates the binomial coefficients used in probability and algebra.

4. Linking permutations and combinations

The fundamental relation:

$^{n} P_{r} =^{n} C_{r} \times r!$

Quantity	Ordered?	Formula	Example
Permutation	Yes	$\frac{n !}{( n - r )!}$	Arranging 3 out of 7 books
Combination	No	$\frac{n !}{r ! ( n - r )!}$	Selecting 3 books for a reading list

This link is frequently tested: students may be asked to derive one from the other.

5. Applications & advantages

Application	Which counting idea?	How it is used
eSewa transaction IDs	Permutations with repetition	10‑digit numeric IDs → $1 0^{10}$ possible codes, ensuring uniqueness.
Daraz order queue	Permutations (linear)	The sequence in which 12 orders are packed determines the daily dispatch schedule; total possible sequences = $12!$ .
Google search result ranking	Combinations (choose top‑k)	From billions of indexed pages, Google selects the best 10 to display → $(10 N)$ possibilities, then orders them (permutation).
Lottery (e.g., Nepal Lottery)	Combinations without repetition	Choose 6 numbers from 1‑45 → $(6 45) = 8, 145, 060$ possible tickets.
Password generation	Permutations with repetition	8‑character alphanumeric password → $6 2^{8}$ possibilities (62 symbols, repetition allowed).

Real‑world worked tie‑in

A bank offers a fixed‑rate loan of NPR 1,000,000 for 5 years. The bank must decide the order in which to approve 5 pending applications (A, B, C, D, E). The profit from each approved loan depends on the order because earlier approvals earn interest sooner. The number of possible approval sequences is $5! = 120$ . By evaluating each permutation, the bank can schedule approvals to maximize total interest earned.

6. Visual summary

flowchart LR
    A["Start: n distinct objects"] --> B["Do we care about order?"]
    B -->|"Yes"| C["Permutation formula nPr"]
    B -->|"No"| D["Combination formula nCr"]
    C --> E["If circular → divide by n"]
    D --> F["If repetition allowed → use stars‑and‑bars"]
    E --> G["Applications: seating, queues"]
    F --> G
    G --> H["Exam: derive, compute, relate"]

7. In the real world

eSewa mobile app – Every payment generates a 12‑digit transaction reference. The reference is a permutation with repetition of digits 0‑9, giving $1 0^{12}$ unique IDs, which prevents duplicate records.
Daraz order fulfillment – On a busy sale day, 12 orders must be packed. The warehouse management system treats the packing list as a permutation; the optimal sequence (minimising travel distance) is chosen from the $12!$ possibilities using heuristics.
Google search – After ranking billions of pages, Google selects the top‑10 to show. This selection is a combination $(10 N)$ ; the displayed order is then a permutation of those 10, influencing click‑through rates.

These examples illustrate that the abstract counting formulas directly control reliability, efficiency, and profitability in services students encounter daily.

8. Common pitfalls & how to avoid them

Pitfall	Why it occurs	Remedy
Mixing up $n P_{r}$ and $n C_{r}$	Forgetting whether order matters	Write a quick note: “order = yes → P, order = no → C”.
Ignoring repetition rules	Assuming all objects are distinct	Check the problem statement for “with/without replacement”.
Forgetting to divide by symmetry in circular arrangements	Treating rotations as distinct	Remember: fix one position as reference, then permute the rest.
Mis‑applying stars‑and‑bars	Using it for distinct objects	Use stars‑and‑bars only when objects are identical and repetitions allowed.

Exam tip

Read the keyword first: “arrange”, “order”, “line up” → permutation; “choose”, “select”, “form a committee” → combination.
Write the generic formula before plugging numbers; this reduces algebraic errors.
For circular problems, explicitly state “fix one object” to justify dividing by $n$ .
When both concepts appear, use the relation $n P_{r} = n C_{r} \times r!$ to switch between them quickly.
Time‑saving shortcut: For small $n$ , sketch a quick tree or use Pascal’s triangle to verify $(r n)$ values.

Google search results page Google SERP displaying top‑10 results (Image: Muhammad Rafizeldi, CC BY 4.0, via Wikimedia Commons)

Based on the TU BCA syllabus for Mathematics I (CAMT104), unit 3.

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