In more detail
Arranging 3 of 10 people in order (first, second and third place) gives 10 × 9 × 8 = 720: ten choices for first place, nine left for second, eight for third. Arranging all n items gives n!, so 8 runners can finish in 8! = 40,320 orders. When repetition is allowed, each position has the full set of choices: nᵏ. When some items are identical, divide by the factorials of the repeat counts.
Why it matters
Permutations count passwords and PINs, race results, seating plans and schedules, and so set the size of a brute-force search. Mixing up "with" and "without repetition" changes the answer: 3-letter codes from 26 letters number 26³ = 17,576 if letters may repeat, but P(26, 3) = 15,600 if they may not.
Example
P(10, 3) = 10! / 7! = 720
4-digit PIN (repeats allowed) = 10⁴ = 10,000
orders of 8 runners = 8! = 40,320
letters of BANANA = 6! / (3! × 2!) = 720 / 12 = 60
BANANA has three A's and two N's; swapping identical letters does not make a new word, hence the division.
Common confusion
Order matters for a permutation; when it does not, use a Combination. Picking a committee of 3 from 10 gives 120 combinations, but assigning president, secretary and treasurer gives 720 permutations.