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Let F q be a finite field with q elements, l an odd prime, and t , v positive integers such that l v | ( q − 1 ) and gcd ( t , l ) = 1 . In this paper, we give a combinatorial result and use it to determine the weight distribution of a cyclic code of length t l m with t | ( q − 1 ) , which is an open question in Yang et al. (2013). Moreover we compute the weight distributions of cyclic codes of length t l m , where q ≡ 3 ( mod 4 ) and t = 4 or 8 . [ABSTRACT FROM AUTHOR]