Permutations (nPr)

{}_{n}P_{r} = \frac{n!}{(n-r)!}

${}_{n}P_{r}$: read 'n permute r': the number of ordered line-ups of r drawn from n
$n!$: factorial: n × (n−1) × … × 1 — all the ways to order the whole pool
$(n-r)!$: dividing by this chops the product off after r slots: only the first r positions matter

Why it must be true

When order matters — assigning distinct roles, ranking finalists, filling 1st/2nd/3rd — count slot by slot: $n$ options for the first slot, $n-1$ for the second, and so on for $r$ slots. The factorial notation $\frac{n!}{(n-r)!}$ just truncates that product cleanly.

Permutations always outnumber combinations by a factor of $r!$ : every unordered group can be arranged $r!$ ways. If swapping two picks changes the outcome, you're counting permutations.

The derivation

Fill $r$ ordered slots from $n$ candidates:

{}_{n}P_{r} = n \times (n-1) \times \dots \times (n-r+1)

Multiply and divide by the remaining factorial to write it compactly:

{}_{n}P_{r} = \frac{n \times (n-1) \times \dots \times 1}{(n-r) \times \dots \times 1} = \frac{n!}{(n-r)!}

When to reach for it

Counting ordered arrangements — rankings, role assignments, sequences where position matters.

Listen for

“ranked first, second, third”“distinct positions / roles / desks”“in how many orders”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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Just multiply r descending terms: 8P3 = 8 × 7 × 6 = 336. No factorials needed.
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Cross-check: permutations = combinations × r!. If 56 and 336 both appear as choices with r = 3, they differ by 6 — order matters here, take the big one.

Traps in applying it

✗Dividing out r! as if order didn't matter — that's the combinations move.
✗Using n^r, which quietly allows the same person to fill two slots.
✗Multiplying n × r instead of the shrinking product n(n−1)…

Limits & criticisms

Assumes no repetition and distinguishable items; arrangements with repeats need the multinomial correction. Factorial growth makes brute-force enumeration hopeless quickly — by n = 15 the orderings exceed a trillion — so large problems are handled with logs and Stirling's approximation.

Where it came from

The factorial's first known rigorous treatment is Levi ben Gerson's 1321 proof that $n$ objects can be ordered in $n!$ ways, with the machinery formalized alongside combinations in Bernoulli's Ars Conjectandi (1713). Permutations are the "order matters" half of classical counting.

Today they count rankings, orderings and assignments — league tables, waterfall priorities, scenario sequences — and serve as the denominator check whenever you must decide whether swapping two picks changes the outcome.

Where it leads

Master this and the following come almost for free:

Combinations (nCr)

Drill this formula Back to the library