Permutations

What the symbols mean

$P_{n,r}$

the number of ordered arrangements, also written $\_n{P}_r$

$n$

the number of items to choose from

$r$

how many are chosen and placed in order (r ≤ n)

$n!$

n factorial: $n\times (n-1)\times \cdots \times 2\times 1$, with $0!=1$

When to use it

Use permutations when different orders count as different outcomes: first, second and third place; officers elected to named posts; arranging books on a shelf. If order does not matter (a hand of cards, a committee), use combinations instead. Repeats allowed (a 4-digit PIN) is simply $n^r$.

Why it works

There are n choices for the first place, $n-1$ for the second, and so on down to $n-r+1$ for the r-th: multiply them. Writing that product as $n!/(n-r)!$ is just a compact way of stopping n! after r factors.

Common mistakes

• Using combinations when order matters: 120 groups of 3, but 720 ordered podiums.
• Computing $n!/r!$ instead of $n!/(n-r)!$.
• Forgetting $0!=1$, so that $P(n,n)=n!$ (all n items in a row).