Permutations and Combinations (Forms 5–6)

Arts 1–4

Foreign Languages 1–4

Introduces counting techniques, factorial notation, and applications in probability.

📘 Topic Summary

Permutations and combinations are fundamental concepts in Additional Mathematics that help us count the number of ways to arrange objects in a specific order or combination.

📖 Glossary

Factorial: The product of all positive integers up to a given number.
Permutation: An arrangement of objects in a specific order.
Combination: A selection of objects without regard to order.
nPr: The number of permutations of n items taken r at a time.
nCr: The number of combinations of n items taken r at a time.

⭐ Key Points

Permutations and combinations are used to solve problems involving arrangements and selections.
Factorial notation is used to simplify calculations involving permutations and combinations.
The order of objects matters in permutations, but not in combinations.
The number of permutations increases rapidly as the number of items increases.
Combinations are often used in probability theory to calculate the likelihood of certain events.

🔍 Subtopics

Introduction to Permutations

A permutation is an arrangement of objects in a specific order. For example, the permutations of the letters A, B, and C are ABC, ACB, BAC, BCA, CAB, CBA. The number of permutations of n distinct objects taken r at a time is denoted by P(n, r) or nPr.

Factorial Notation

The factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are used to calculate permutations and combinations.

Permutation Formulas

The formula for calculating the number of permutations of n distinct objects taken r at a time is P(n, r) = n! / (n-r)! The formula can be rearranged as P(n, r) = n × (n-1) × ... × (n-r+1).

Introduction to Combinations

A combination is a selection of objects without regard to order. For example, the combinations of the letters A, B, and C are ABC, ACB, AB, BC, CA, CB, AB, AC, BC. The number of combinations of n distinct objects taken r at a time is denoted by C(n, r) or nCr.

Combination Formulas

The formula for calculating the number of combinations of n distinct objects taken r at a time is C(n, r) = P(n, r) / r!. The formula can be rearranged as C(n, r) = n! / (r!(n-r)!).

Real-World Applications

Permutations and combinations have many real-world applications. For example, they are used in cryptography to encode secret messages, in computer programming to generate random numbers, and in statistics to analyze data.

Common Mistakes to Avoid

When calculating permutations or combinations, it is easy to make mistakes by forgetting to divide by the factorial of r. Always double-check your calculations to ensure accuracy.

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