multiplication rule
arrangements (Permutationen)
Sorting 5 people on to 5 chairs in order. $${5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 60}$$
Also, written as ${5!}$ - “5 factorial”
different arrangements (Permutationen in German)
${n!}$ is n factorial
sitting in a circle at the “Round Table”
Here, each arrangement can be shifted in one direction n times without changing the order.
So, 5 people around a round table is $${\frac{5!}{5} = 4! = 24}$$
permutations (Variationen)
Sorting 3 out of 7 people in order
$${7 \cdot 6 \cdot 5 = 210}$$
$${7 \cdot 6 \cdot 5 = \frac{7 \cdot 6 \cdot 5 \cdot \color{orange}{4 \cdot 3 \cdot 2 \cdot 1}}{\color{orange}{4 \cdot 3 \cdot 2 \cdot 1}}= \frac{7!}{4!} = {}^{7}P_{4}}$$
generally
$${{}^{n}P_{r} =\frac{n!}{(n-r)!} }$$
combinations (Kombinationen)
similar to permutations but where the order does not matter.
so we need to divide by the number off possible arrangements of selected items:
$${{}^{n}C_{r} =\frac{n!}{(n-r)! \cdot r!} = \binom n r}$$
“selecting” 3 out of 5 - not considering the order.
$${\frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(2 \cdot 1) \cdot (3 \cdot 2 \cdot 1)}= \frac{5!}{(5-3)!3!} = {}^{5}C_{3} = \binom 5 3}$$
Note
$${{}^{n}C_{r} =\frac{{}^{n}P_{r}}{r!} }$$
$\binom n r$ is also known as the binomial coefficient (der Binomialkoeffizient) in the expansion of $(a + b)^2$. Pascal’s triangle is a quick way to find the coefficients for small numbers of n.
selecting a specific numbers of different items from 2 groups
Forming a commitee with 3 members out of 4 men and 3 women.
$${\binom 7 3 = 35}$$ combinations of selecting 3 out of 7.
Forming a commitee with exactly 2 men and 1 woman.
males: ${\binom 4 2 = 6}$ - females: ${\binom 3 1 = 3}$
$${\binom 4 2 \cdot \binom 3 1 = 6 \cdot 3 = 18}$$
possible ways.
Pascal’s triangle
table of formulas
TODO