Permutations and combinations

 

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics that deal with arranging and selecting objects.

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1. Permutations (Order Matters)

A permutation is an arrangement of objects in a specific order.

Formula for Permutations

The number of ways to arrange $r$ objects from a set of $n$ distinct objects:

$$P(n, r) = \frac{n!}{(n-r)!}$$

Example of Permutation

How many ways can 3 people (A, B, and C) be arranged in 2 seats?

$$P(3,2) = \frac{3!}{(3-2)!} = \frac{3!}{1!} = \frac{3 \times 2 \times 1}{1} = 6$$

Arrangements: (AB, BA, AC, CA, BC, CB)

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2. Combinations (Order Doesn’t Matter)

A combination is a selection of objects where order does not matter.

Formula for Combinations

The number of ways to choose $r$ objects from $n$ distinct objects:

$$C(n, r) = \frac{n!}{r!(n-r)!}$$

Example of Combination

How many ways can you choose 2 people from a group of 3 (A, B, and C), ignoring order?

$$C(3,2) = \frac{3!}{2!(3-2)!} = \frac{3!}{2! \times 1!} = \frac{3 \times 2 \times 1}{(2 \times 1) \times 1} = 3$$

Selections: (AB, AC, BC)

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Key Differences Between Permutations and Combinations

Feature	Permutations	Combinations
Order	Matters	Doesn't matter
Formula	$P(n, r) = \frac{n!}{(n-r)!}$	$C(n, r) = \frac{n!}{r!(n-r)!}$
Example	Arranging books on a shelf	Selecting a committee from a group