The main contribution of this work is to present several hardware implementations of an "n choose k" counter (C(n,k) counter for short), which lists all n-bit numbers with (n-k) 0's and k 1's, and to show their applications. We first present concepts of C(n,k) counters and their efficient implementations on an FPGA. We then go on to evaluate their performance in terms of the number of used slices and the clock frequency for the Xilinx VirtexII family FPGA XC2V3000-4. As one of the real life applications, we use a C(n,k) counter to accelerate a digital halftoning method that generates a binary image reproducing an original gray-scale image. This method repeatedly replaces an image pattern in small square regions of a binary image by the best one. By the partial exhaustive search using a C(n,k) counter we succeeded in accelerating the task of finding the best image pattern and achieved a speedup factor of more than 2.5 over the simple exhaustive search.