What is the fundamental counting principle for arranging n distinct objects?

Q: What is the fundamental counting principle for arranging n distinct objects?

For n distinct objects, successive choices are n, n−1, …, 1, giving n!. The counting rule needs the stated number of choices after each preceding choice; this is not a probability-independence assumption.

For n distinct objects, successive choices are n,

n-1

, …, 1, giving n!. The counting rule needs the stated number of choices after each preceding choice; this is not a probability-independence assumption.

Conditions

Objects are distinct.
Order matters.

Reasoning, step by step

Identify the number of options for the first position (n).
Identify the number of remaining options for the second position (n-1).
Continue this process until the last position (1).
Multiply the number of choices for each position together.
Recognize the product as n!.

Example

In the video, for the word RATTATA with 7 letters, the presenter calculates the total arrangements assuming all letters are unique by multiplying $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ , which equals $7!$ .

Common misconceptions

Thinking that the fundamental counting principle implies probability independence.
Forgetting that the number of choices decreases by one for each subsequent position.

Watch the explanation

XHow many permutations are there of the letters of the word Rattata?

0:13 – 0:36Open original video ↗

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Connected concepts

Combinatorial counting →

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Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.