Abstract The conditional maximum likelihood estimator of the fixed-effect logit model suffers from a curse of dimensionality that may have severely limited its use in practice. As the number of alternatives and the number of choice situations per individual increase, the number of addends in the denominator of the fixed-effect logit formula grows exponentially. We propose to by-pass this curse of dimensionality by exploiting a classic result by McFadden (1978) and to consistently estimate the fixed-effect logit model on random samples of permutations of the observed choice sequences.