A combination of mathematical formulas commonly used summary - Update

• White concluded in error please correct me big brother

Permutations

arrangement

1. No other restrictions, select the r out from the n object species all permutations case \ (A (^ R_n) = \ FRAC {n!} {(NR)!} \) R> when n \ (A (^ r_n) = 0 \)

2. Select r objects from the n th species circular arrangement of \ (P (^ r_n) = \ frac {A (^ r_n)} {r} \)

Arrangement of multiple sets

1. Each set of n mutually different elements, each element has \ (\ infty \) species (infinite multiple set), arranged to take r a species in which n \ (n ^ r \)

2. Each set of n mutually different elements, each element has a \ (a_1, a_2, a_3 ... a_n \) Species (multiple finite set), a take r n in this species, when \ (min ({ a_1, a_2, ... a_n}) > = r \) , the number of permutations is still \ (n ^ r \)

3. Each set of n mutually different elements, each element has a \ (a_1, a_2, a_3 ... a_n \) Species (multiple finite set), which is a full permutation \ (\ frac {(a_1 + a_2 + a_3 + ... + a_n)!} {{ a_1}! {a_2}! ... {a_n}!} \)

4. Each set of n mutually different elements, each element has a \ (a_1, a_2, a_3 ... a_n \) Species (multiple finite set), a take r n in this species, when \ (min ({ a_1, a_2, ... a_n}) <r \) when arranged as \ (\ frac {r!} {r {a_1}! {a_2}! ... {a_n}!} \)

combination

1. Unrestricted, selected from among the n objects is a combination of the r objects \ (C (n, r) = \ n {FRAC!} {R! (NR)!} \) , Also written \ ((^ n_r) = \ {n-FRAC!} {R & lt! (NR)!} \) , R & lt> when n-, \ (C (n-, R & lt) = 0 \)

A combination of multiple sets

1. Each set of n mutually different elements, each element has \ (\ infty \) species (infinite multiple set), to take a combination of r in this species is n \ ((^ {n + r -1} _ {r}) = (^ {n + r-1} {n-1}) \)

2. Each set of n mutually different elements, each element has a \ (a_1, a_2, a_3 ... a_n \) Species (multiple finite set), a take r n in this species, when \ (min ({ a_1, a_2, ... a_n}) > = r \) , the number of combinations of \ ((^ {n + r -1} _ {r}) = (^ {n + r-1} {n-1 }) \)

3. Each set of n mutually different elements, each element has a \ (a_1, a_2, a_3 ... a_n \) Species (multiple finite set), a take r n in this species, when \ (min ({ a_1, a_2, ... a_n}) <r \) , the combination $$

Binomial theorem

• \((a+b)^n=\sum_0^nC(_n^i)a^ib^{n-i}\)

Pigeonhole principle

• n + 1 n pigeon flying pigeonholes, there must be two pigeons fly the same pigeonholes

Generating function articles

\((1-x)^{-m}=\sum_0^\infty{x^i(^{m+i-1}_{m-1})}\)