Chemical Kinetics- Net Number of Particles
Actually, the rate law for the species $\ce{C}$ is written as:

$$\frac{\mathrm{d}\ce{C}}{\mathrm{d}t} = k \ce{A} \ce{B} - k' \ce{C} \ce{y}$$

If you integrate this, you would get an equation in terms of the number of particles, as you require. However, this is a very typical integral, and there would be a number of other arbitrary constants, which would make the final result not equal to the one you've shown.

What you've written is a mistake on replacing the rate of formation of $\ce{C}$ with its number of particles itself. Both are not equivalent.