= Solution
The monopole vanishes exactly when
$$
\boxed{N=M+1},
$$
and the dipole vanishes exactly when
$$
\boxed{M=1}.
$$
Since $N$ and $M$ are positive integers, both vanish precisely for
$$
\boxed{N=2,\qquad M=1}.
$$
Assume now that the monopole has been cancelled. If $Qd$ is held fixed while $d\to0$, the quadrupole coefficient $Qd^2=(Qd)d$ tends to zero, whereas the dipole coefficient $(M-1)Qd$ has a finite limit. For $M\ne1$ this is a point-dipole limit; for $M=1$ the limiting external potential vanishes.
If instead $Qd^2$ is held fixed, the dipole coefficient is
$$
(M-1)Qd=(M-1)\frac{Qd^2}{d}.
$$
It diverges unless $M=1$. When $M=1$ and $N=2$, both lower multipoles vanish and the quadrupole coefficient has a finite nonzero limit. These are the <dipole and quadrupole scaling limits of three collinear charges>.
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