Basic Understanding of Convolution Operation I would really appreciate any help in understanding this.

If $$y[n] = x[n]*r[n]\tag{1}$$ then what is $y[n-n_0]$ for any system?

Is it $$y[n-n_0] = x[n-n_0]*r[n-n_0]\tag{2}$$ 
or    $$y[n-n_0] = x[n-n_0]*r[n]\tag{3}$$ 
or    $$y[n-n_0] = x[n]*r[n-n_0]\tag{4}$$ 

Will that result (answer to the above) change when we now have an LTI system with $h[n]=r[n]$ being its impulse response?

Thank you very much
Since $y[n-n_0] = y[n] \ast \delta [n - n_0]$ and convolution is both associative and distributive, is it not the case that

$$
y[n-n_0] = \left(x[n] \ast r[n] \right) \ast \delta[n-n_0] = x[n] \ast r[n-n_0] = x[n-n_0] \ast r[n]?
$$