The highest order number that can be observed with a grating is determined by the equation:
mλ = d(sin θ + sin θi)
where m is the order number, λ is the wavelength of light, d is the spacing between the lines on the grating, θ is the angle of incidence, and θi is the angle of diffraction.
Given that the grating has 1555 lines/cm, and the wavelength of light is 565 nm, we can use this equation to find the highest order number:
mλ = d(sin θ + sin θi)
m = d / λ
m = 1555 lines/cm / (565 nm/1cm)
m = 2.76
The highest order number that can be observed with this grating is 2.76. It's noteworthy that the order number is a positive number and it will not be a whole number, this means that it's not possible to observe the order number exactly 2.76 but it's the highest possible number that can be observed.