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Multiplicity fluctuations of positively, negatively and all charged hadrons in the forward hemisphere were studied in central Pb+Pb collisions at 20A, 30A, 40A, 80A and 158A GeV. The multiplicity distributions and their scaled variances $\omega$ are presented in dependence of collision energy as well as of rapidity and transverse momentum. The distributions have bell-like shape and their scaled variances are in the range from 0.8 to 1.2 without any significant structure in their energy dependence. No indication of the critical point fluctuations are observed. The string-hadronic model UrQMD significantly overpredicts the mean, but approximately reproduces the scaled variance of the multiplicity distributions. The predictions of the statistical hadron-resonance gas model obtained within the grand-canonical and canonical ensembles disagree with the measured scaled variances. The narrower than Poissonian multiplicity fluctuations measured in numerous cases may be explained by the impact of conservation laws on fluctuations in relativistic gases.