Multinomial Coefficients: Binomial Products and Counting

In the setting of Finite Combinatorics: Notation, let $m\in\mathbb{N}$ and $k_{1},\dots,k_{m}\in\mathbb{N}_{0}$.

(i) Adding an entry. For every $k_{m+1}\in\mathbb{N}_{0}$,

$$\binom{k_{1}+\dots+k_{m+1}}{k_{1},\dots,k_{m+1}}=\binom{k_{1}+\dots+k_{m}}{k_{1},\dots,k_{m}}\binom{k_{1}+\dots+k_{m+1}}{k_{m+1}}.$$

(ii) Two entries. For all $a,b\in\mathbb{N}_{0}$,

$$\binom{a+b}{a,b}=\binom{a+b}{a}.$$

(iii) Counting maps by fibre sizes. Let $n=k_{1}+\dots+k_{m}$. Then

$$\#\{\varphi\in[m]^{[n]}:\#\varphi^{-1}\{i\}=k_{i}\text{ for every }i\in[m]\}=\binom{n}{k_{1},\dots,k_{m}}.$$