Combinations

At a glance 12 formulas

1. 1Selections${}^{n}C_{r} = \dfrac{n!}{r!\,(n - r)!}$$r \text{ chosen from } n;\ {}^{n}C_{r} = {}^{n}C_{n - r}$
2. 2Select, then arrange${}^{n}P_{r} = {}^{n}C_{r} \times r!$$\text{pick the } r \text{ from } n \text{ first, then order them}$
3. 3Always in, always out$k \text{ always in: } {}^{n - k}C_{r - k}$$k \text{ always out: } {}^{n - k}C_{r};\ r \text{ chosen from } n$
4. 4From two groups${}^{a}C_{x} \times {}^{b}C_{y}$$x \text{ from a group of } a,\ y \text{ from a group of } b$
5. 5At least kFor 'at least k', add the cases k, k + 1 and so on; if those are many, take all selections minus the few that fail. Any number, at least one, from $n$ different things: $2^n - 1$.
6. 6A pair never together${}^{n}C_{r} - {}^{n - 2}C_{r - 2}$$r \text{ from } n;\ \text{two named people never together}$
7. 7Splitting into groups$\dfrac{(a + b + c)!}{a!\,b!\,c!}$$\text{sizes } a, b, c;\ k \text{ equal unnamed groups: } \div\, k!$
8. 8Nobody left outDifferent items to k people, each getting at least one: split the items into k non-empty bundles, then hand them out in $k!$ ways. 4 prizes, 3 people: $6 \times 3! = 36$.
9. 9Handshakes and diagonals${}^{n}C_{2} = \dfrac{n(n - 1)}{2}$$n \text{ people};\ n\text{-gon diagonals: } \dfrac{n(n - 3)}{2}$
10. 10Collinear points$\text{triangles} = {}^{n}C_{3} - {}^{m}C_{3}$$m \text{ of the } n \text{ collinear};\ \text{lines} = {}^{n}C_{2} - {}^{m}C_{2} + 1$
11. 11Rectangles on a grid${}^{m + 1}C_{2} \times {}^{n + 1}C_{2}$$m \times n \text{ grid};\ \text{squares: } mn + (m - 1)(n - 1) + \cdots$
12. 12Triangles from a polygonFrom the $n$ corners of a polygon: ${}^{n}C_{3}$ triangles, of which $n$ share two sides with it and $n(n - 4)$ share exactly one.