= Solution
For normal incidence on a <diffraction grating>, adjacent grooves contribute a path difference $d\sin\theta$. <Constructive interference> occurs when the <grating equation> $d\sin\theta=m\lambda$ holds. Thus the groove spacing is
$$
d=\frac{3(589\times10^{-9}\,\mathrm m)}{\sin32^\circ}
=3.334\times10^{-6}\,\mathrm m,
\qquad
\boxed{N=\frac{10^{-3}\,\mathrm m}{d}=299.9\ \text{lines per mm}\simeq300\ \text{lines per mm}.}
$$
An allowed propagating <diffraction order> must have $|\sin\theta|\leq1$. Therefore
$$
|m|\leq\frac d\lambda=\frac3{\sin32^\circ}=5.6615,
\qquad
\boxed{|m|_{\max}=5.}
$$
The fifth <diffraction order> is the highest geometrically permitted order for this <wavelength>. Its actual brightness depends on groove shape and the <single-slit diffraction> envelope, which the source does not specify; geometric permission alone does not guarantee a bright observable maximum.
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