Light with a wavelength of about 510 nm is made to pass through a diffraction grating. the angle formed between the path of the incident light and the diffracted light is 12.0° and forms a first-order bright band.

the number of lines per mm in the diffraction grating, rounded to the nearest whole number, is lines per mm.

formula? ?

First, we use the relationship between the spacing of the openings, the wavelength of light, the order of the maxima or minima and the angle of diffraction which is:
d*sinθ = nλ, where d is the spacing, θ is the angle, n is the order and λ is the wavelength. We will use the wavelength in mm so that we may obtain the spacing in mm (510 x 10⁻⁶ mm). Solving for d,
d = (1)(510 x 10⁻⁶) / sin(12)
d = 2.45 x 10⁻³ mm
Using this, we may calculate openings per mm,
1 / 2.45 x 10⁻³ 
= 408 lines per mm.