Permutations

At a glance 12 formulas

1. 1Multiply, then addFill the places one at a time and multiply the choices; add only separate cases that cannot both happen.
2. 2Arrangements${}^{n}P_{r} = \dfrac{n!}{(n - r)!}$$r \text{ places from } n \text{ different things};\ {}^{n}P_{n} = n!$
3. 3Restricted place firstA digit or a seat with a condition? Fill it first, then arrange the rest. 3-digit even numbers using the digits 1 to 5, no repeats: $2 \times 4 \times 3 = 24$.
4. 4Repeats allowed$\text{repeats allowed: } n^{r}$$r \text{ places, any of } n \text{ things in each}$
5. 5Zero cannot leadIf 0 is one of the digits, fill the first place from the non-zero digits, or count everything and take away the numbers that start with 0.
6. 6Identical items$\dfrac{n!}{p!\,q!}$$n \text{ items: } p \text{ alike of one kind, } q \text{ of another}$
7. 7Grid routes$\dfrac{(a + b)!}{a!\,b!}$$a \text{ steps right and } b \text{ steps up, in any order}$
8. 8Together: glue them$(n - k + 1)! \times k!$$k \text{ of } n \text{ glued};\ \text{a pair apart: } n! - 2(n - 1)!$
9. 9Keep apart: use gaps$n \text{ items} \Rightarrow n + 1 \text{ gaps}$$\text{4 boys, 3 girls, no two girls together: } 4! \times {}^{5}P_{3}$
10. 10Fix one seat$\text{round table: } (n - 1)!$$n \text{ people};\ \text{necklace (can flip): } \tfrac{(n - 1)!}{2}$
11. 11Fixed order$\dfrac{n!}{k!}$$k \text{ of the } n \text{ items must keep a set order}$
12. 12Dictionary rankPlace by place, try each different unused letter that comes earlier in the alphabet and count the arrangements of the rest, dividing for repeats. Add them all, then add 1.