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Counting: permutations and combinationsStatistics and probabilityHigh schoolabout 0:35

How many ways can you arrange 5 of 5 items in order?: animated lesson

A narrated, step-by-step counting: permutations and combinations video lesson with captions. Final answer: P(5, 5) = 120.

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Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.2 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
How many ways can you arrange 5 of 5 items in order?
2. Step by step
1. $P(5, 5) = \frac{5!}{(5 - 5)!} = \frac{5!}{0!}$
  Use the permutation formula
2. $5 \times 4 \times 3 \times 2 \times 1 = 120$
  Cancel 0! to leave the first 5 factors of 5!
3. The answer
$120$
120
4. Check
- Check: P(5, 5) = C(5, 5) × 5! = 1 × 120 = 120.

Transcript

The problem. Here is the problem: How many ways can you arrange 5 of 5 items in order? Permutations nPr (order matters).

Step by step. Use the permutation formula. Cancel 0! to leave the first 5 factors of 5!

The answer. So the answer is P(5, 5) equals 120. There are 120 ordered arrangements of 5 items chosen from 5.

Check. Check: P(5, 5) equals C(5, 5) times 5! equals 1 times 120 equals 120.

About counting: permutations and combinations

Count arrangements and selections with permutations and combinations.

How many ways can you arrange 5 of 5 items in order?