Path difference = n*lambda = d*sin(theta)

where n Is some positive real number, d is the inter-slit distance, theta is the angle is subtended by the fringe at the point lying midway the two slits and lambda is the wavelength of light.

For the very last fringe (bright) on the distant screen, the angle theta = 90 degrees. So the above formula becomes n*lambda = d*sin(90) = d*1 = d

Therefore, n*lambda = d which implies n = d/lambda

However, because the number of fringes has to be a whole number, it becomes necessary to introduce a floor function (aka greatest integer function). That is,

floor(n) = [n] = number of fringes on one half of the screen

Hence, total number of fringes = number of fringes on both halves of the screen plus the lone central maxima at the center of the screen

That is, total number of fringes = 2[n]+ 1 = 2[d/lambda]+1

This is the method of calculating total number of fringes based on the conventional analysis of Young’s Double Slit Experiment

Hope it helps

Thanks and regards

Arun