Permutation and combination full tutorials last part : for Class 12

Permutation & Combination

Grade 12 · CDC Nepal · By Ghanashyam Adhikari

Chapter 1 · Mathematics · Grade 12

How many ways can it happen?

A hands-on tour of Permutation and Combination — the mathematics of arranging and selecting. Work through the theory, watch the animations respond to your input, and test yourself with locked exercises and a full CDC-model exam.

• 01 Distinguish arrangement (order matters) from selection (order doesn't).
• 02 Derive and apply ⁿP_r and ⁿC_r.
• 03 Handle repetition, restriction, and circular cases.
• 04 Solve real committee, seating and geometry problems.
• 05 Sit a full CDC-model examination and get instant feedback.

Section

Theory & Formula

Every definition, derivation and worked example from the chapter, fully rendered.

Section

Practice

20 MCQs plus 20 password-protected worked exercises across two problem sets.

Section

Examination

Generate a fresh 15-question CDC-model paper and get an instant score report.

Background

History of Permutation and Combination

Counting arrangements is one of the oldest branches of mathematics — from Sanskrit poetry meters to modern data science.

Applications

Real-Life Applications

Applications of permutations and combinations, drawn from everyday situations.

🔐

Passwords & Security

In cybersecurity and encryption, we often rely on permutations and combinations to create strong passwords and secure systems. When you set a password, you select characters from a set of possibilities, such as letters, numbers, and symbols. The more options you have for each character position, the more potential combinations there are for your password.

🎟️

Lottery & Gambling

Permutations and combinations are important in lottery games and other gambling activities. In the lottery, winning is just a permutation of numbers picked from the pool of numbers. It is important to get the idea of the probabilities of each possible combination because it can influence the decisions players make, however luck is also an inseparable element.

🪑

Seating Arrangements & Event Planning

A seating plan is a very important element of the planning process regardless of whether you are organizing a theater performance, a wedding reception, or a conference. Every single individual attending the event is directly impacted by how well the seating system works. The number of possibilities created by permutations and combinations makes it easier for planners to bring each guest to the exact spot where they belong.

Interactive

Interactive Visualization

Scenario: The Pizza Shop and the Locker. Imagine two tasks — entering a 3-digit pin into a bicycle padlock, and picking 3 toppings from 5 available choices (Tomato, Onion, Corn, Olives, Mushroom).

Permutation — order matters

The real-life task: entering the code 4-2-9 into your padlock. If you enter 9-2-4, the lock will not open — changing the position of the exact same numbers creates a totally different outcome. Tap the keys to fill the three slots and test it.

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Combination — order doesn't matter

Pick any 3 toppings for your bowl, in any order. Whether the chef adds Tomato first or Corn first, the final bowl is identical.

Your bowl is empty — pick 3 toppings.

Draw tokens: Arrangement vs Group

Click numbered tokens from the pool. In Arrangement mode each click fills the next ordered slot (a permutation). In Group mode tokens just join an unordered set (a combination) — the same tokens in a different click-order still form the identical group.

Token pool

Ordered slots

Theory

Definitions

Definition of Permutation

A permutation is an arrangement of objects in a specific order. In a permutation, the order of selection is important.

Example: Suppose there are three letters A, B and C. The different permutations of taking two letters are AB, BA, AC, CA, BC, CB — there are 6 different permutations, because changing the order creates a new arrangement. \( {}^{6}P_{3} = \dfrac{6!}{(6-3)!} = 120 \)

🧮 Try it yourself

Pick a type, enter your own numbers, and calculate instantly.

Choose type Linear Permutation (nPr) Circular Permutation Permutation — Not All Distinct Combination (nCr)

n (total objects)

r (chosen)

Circle type Case A — directional (table) Case B — non-directional (necklace)

Repeated group sizes (comma-separated)

Kinds of Permutation

1. Linear Permutation of Distinct Objects

When we arrange r objects taken from a set of n distinct objects in a straight line, with no repetition.

\[ {}^{n}P_{r} = \dfrac{n!}{(n-r)!} \]

n = total items, r = items selected, \(n! = n\times(n-1)\times(n-2)\times\cdots\times1\)

Example 1. In how many ways can 3 books be chosen and arranged on a shelf out of 5 different books?
\( {}^{5}P_{3} = \dfrac{5!}{(5-3)!} = \dfrac{5!}{2!} = 5\times4\times3 = 60 \). ∴ 60 ways.

Example 2. How many 4-digit numbers can be formed using 1,2,3,4,5 without repeating any digit?
\( {}^{5}P_{4} = \dfrac{5!}{1!} = 120 \). ∴ 120 different 4-digit numbers.

2. Permutation of Objects Not All Distinct

When the n objects are not all distinct, i.e. some items repeat.

\[ \text{Arrangements} = \dfrac{n!}{n_1!\,n_2!\,\cdots\,n_k!} \]

where \(n_1, n_2,\dots\) are the frequencies of identical items.

Example 1. Arrangements of the letters of "MATH" (all distinct, n = 4): \(4! = 24\).

Example 2. Arrangements of the letters of "BOOK" (O repeats twice): \(\dfrac{4!}{2!} = 12\).

3. Circular Permutation

When objects are arranged around a closed circle instead of a straight line.

• Case A (directional — seating around a table): arranging n distinct objects in a circle is \((n-1)!\).
• Case B (non-directional — necklaces, garlands, beads): when clockwise and anticlockwise look identical, divide by 2: \(\dfrac{(n-1)!}{2}\).

\[ \text{Case A: } (n-1)! \]

\[ \text{Case B: } \dfrac{(n-1)!}{2} \]

Example 1. 5 people seated around a circular table: \((5-1)! = 4! = 24\) ways.

Example 2. 4 different beads strung into a necklace: \(\dfrac{(4-1)!}{2} = \dfrac{3!}{2} = 3\) ways.

Definition of Combination

A combination is a selection of objects without considering their order. In a combination, the order of selection is not important.

\[ {}^{n}C_{r} = \dbinom{n}{r} = \dfrac{n!}{r!\,(n-r)!} \]

Example: With letters A, B, C, the combinations of taking two letters are AB, AC, BC — AB and BA are the same combination, so there are only 3 different combinations. \( \dbinom{7}{4} = \dfrac{7!}{4!(7-4)!} = 35 \)

Key Difference

Permutation	Combination
Arrangement of objects	Selection of objects
Order is important	Order is not important
Example: AB ≠ BA	Example: AB = BA

Theory

Formula & Derivation

Practice

Quizzes

Class 12 Mathematics — Permutation and Combination. 20 multiple choice questions: 10 Knowledge, 5 Understanding, 5 Application.

Score: 0 / 20

Practice · Part 1

Exercise 1 — Permutations

10 question types covering the fundamental counting principle through circular permutations. Solutions are password-protected.

Practice · Part 2

Exercise 2 — Combinations

10 question types covering committee selection through geometric applications. Solutions are password-protected.

Assessment

Examination — CDC Nepal Model

5 questions of 2 marks, 5 of 3 marks and 5 of 5 marks, drawn from across the chapter and varying in cognitive level. Answer using the equation editor, the drawing pad, or by uploading an image, then submit for an instant score report.

Total: — marks

Result

0Marks obtained

0Total marks

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Strengths

To improve

Auto-marking checks your final answer against the answer key. Steps and reasoning aren't graded automatically — after submitting, compare your working against the model solutions in Exercise 1 & 2.

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