Answer
$\rm3 \;bright \;fringes.$
Work Step by Step
To find the number of bright fringes, we need to find $y_1$, $y_2$, etc to see if it is fit on the 2 m-wide screen or not.
In a diffraction grating,
$$y_m=L\tan\theta_m$$
and
$$d\sin\theta_m=m\lambda$$
The maximum distance of $y_m$ here is only 1 m since there is a maximum bright firing in the center of the screen and the first order of bright fringes are left and right (or up and down) of the center maximum fringe.
We need to solve for $m$ to see what is the result of it when $y_m=1$ m. If it is an integer number, so there are two bright fringes on the two edges of the screen. And if not, then we need to go to the integer number less than the result.
$$m=\dfrac{d\sin\theta_m}{\lambda}$$
where $\theta=\tan^{-1}\left[ \dfrac{y_m}{L}\right]$
$$m=\dfrac{d\sin\left( \tan^{-1}\left[ \dfrac{y_m}{L}\right]\right)}{\lambda}$$
Plugging the known, where $y_1=1$ m, $d=\frac{1}{500} $ mm, $L=2$ m, and $\lambda=510$ nm
$$m=\dfrac{\frac{1}{500}\times 10^{-3}\sin\left( \tan^{-1}\left[ \dfrac{1}{2}\right]\right)}{510\times 10^{-9}}=\bf 1.75$$
Since $m=1.75$, the screen can't contain $m=2$ bright fringes.
And hence, we only got three bright fringes, the central maximum one, and two bright fringes on each side of the central one.
$$\boxed{N_{\rm bright\;fringes}=\color{red}{\bf 3}}$$
