Recently in my chemistry class I learned about reaction orders. For some reaction $\ce{r_1A1}+\ce{r_2A2}+\dots+\ce{r_nA_n}\rightarrow\ce{B}$, $m_1,\dots{,m_n}$ are the orders of that reaction. The meaning of the order $m_i$ is as follows: If we increase the concentration of reactant $\ce{A_i}$ by a factor $\lambda$ we know that the reaction rate increases by a factor of $\lambda^{m_i}$.
In general, I learned, the orders must be determined experimentally. However, for reactions without intermediate steps, they coincide with the coefficients of their respective reactant. I might've found a way to derive this fact, which is why I'd like to ask whether my derivation is valid. Moreover, in my opinion, this is a rather lengthy derivation, which is why I'd also like to ask whether there is an easier one.
Here we go:
My idea is to show that the reaction rate $v$ in fact increases by approximately $\lambda^{r_i}$ if we increase the amount of $\ce{A_i}$ by $\lambda$.
The number of possible collisions $C$ of $r_1, \dots, r_n$ particles from substances $\ce{A1},\dots, \ce{A_n}$ which can cause a reaction, is proportional to ${N_i\choose r_i}$, where $N_i$ is the number of particles of $\ce{A_i}$. Now see what happens if we increase the amount of $\ce{A1}$ by $\lambda$: For the binomial coefficient we get
$$ {\lambda N_i\choose r_i} = \lambda N_i(\lambda N_i-1)\dots(\lambda N_i-r_i+1)\\ \approx \lambda^{r_i} N_i(N_i-1)\dots(N_i-r_i+1)\\ = \lambda^{r_i} {N_i\choose r_i}, $$
so $C$ also increases by $\lambda^{r_i}$. And because $C$ is itself proportional to the reaction rate $v$, $v$ also increases by the same factor and we are done.
