AI Tutor
Imagine eight sprinters line up for an Olympic 100-meter final, but only three can stand on the medal podium. Does finishing Gold-Silver-Bronze feel the same as Bronze-Silver-Gold?
Clearly, position matters. Whenever the sequence or arrangement of items creates a distinct outcome, we call that arrangement a permutation.
Before we calculate podium finishes, what happens when we arrange every single item in a group?

Arranging All Items: Factorials

If you have 4 books to line up on a shelf, you have 4 choices for the first spot, 3 choices left for the second, 2 for the third, and only 1 choice for the last.
Multiplying these sequential choices gives 4×3×2×1=244 \times 3 \times 2 \times 1 = 24 total arrangements. We write this descending product compactly using factorial notation, represented by an exclamation point (4!4!).
French mathematician Christian Kramp introduced the exclamation mark notation n!n! in 1808 simply because printers needed an easy shorthand for long product chains.
In general, the number of ways to arrange all nn distinct objects in a line is n!=n×(n−1)×⋯×2×1n! = n \times (n - 1) \times \dots \times 2 \times 1.
What happens if you have 8 runners, but you only have 3 medal spots to fill?

Selecting and Arranging: The nPr Formula

We start filling podium spots from left to right: there are 8 possible Gold medalists, 7 remaining candidates for Silver, and 6 left for Bronze.
That gives 8×7×6=3368 \times 7 \times 6 = 336 possible podium orders. Notice that 8×7×68 \times 7 \times 6 looks just like 8!8!, except the tail end (5×4×3×2×1=5!5 \times 4 \times 3 \times 2 \times 1 = 5!) has been chopped off.