Rate of reaction for hundred reactants
The questions has at least two aspects to address.

The first thing is, there is no guarantee that the reaction is first-order with respect to each of the reactants in first place. In other words, without further investigation you can not just take $r=k[\ce{A}][\ce{B}]$ because $r=k[\ce{A}][\ce{B}]^2$ or $r=[\ce{A}]^2[\ce{B}]$ or something more exotic may happen, depending on the nature of the reaction.

OK. So let's say for simplicity we take $r=k[\ce{A}][\ce{B}]$ when the system has only $\ce{A}$ and $\ce{B}$, as we don't need to be quite educationally correct. Then yes, after mixing many reactants, the equation still holds for each pair of the reactants, provided they are elementary reactions. It will be a completely different story if, for example, neither of two of the reactions are elementary and they share the same reaction intermediates. 

I also have a reminder that when stepping from the expression of the rate to the expression of concentration with respect to time, you have to be careful and do the integration properly instead of just naively extrapolating the concentration expression when only two reactants exists. Let's say that we don't consider chemical equilibrium for the moment (i.e. consider all the reverse reactions to be prohibitively difficult to happen), then as far as I can see, the $r-t$ curve will not be smooth throughout the time and will have some turning point every time a reactant is fully consumed.

Side note: I suggest that you do your formatting properly in your program for the superscripts and subscripts.