Binary primitive triple-error-correcting Bose-Chaudhuri-Hocquenghem (BCH) codes of length n=2m-1 have been the object of intensive studies for several decades. In the 1970s, their covering radius was determined in a series of papers to be ρ=5. However, one problem for these codes that has been open up to now is to find their coset distribution. In this paper this problem is solved and the number of cosets of each weight in any binary primitive triple-error-correcting BCH code is determined. As a consequence this also gives the coset distribution of the extended codes of length N=2m with minimum distance 8.