The impact of selection bias on the results of clinical trials has been analyzed extensively for trials of two treatments, yet its impact in multi-arm trials is still unknown. In this paper, we investigate selection bias in multi-arm trials by its impact on the type I error probability. We propose two models for selection bias, so-called biasing policies, that both extend the classic guessing strategy by Blackwell and Hodges. We derive the distribution of the F-test statistic under the misspecified outcome model and provide a formula for the type I error probability under selection bias. We apply the presented approach to quantify the influence of selection bias in multi-arm trials with increasing number of treatment groups using a permuted block design for different assumptions and different biasing strategies. Our results confirm previous findings that smaller block sizes lead to a higher proportion of sequences with inflated type I error probability. Astonishingly, our results also show that the proportion of sequences with inflated type I error probability remains constant when the number of treatment groups is increased. Realizing that the impact of selection bias cannot be completely eliminated, we propose a bias adjusted statistical model and show that the power of the statistical test is only slightly deflated for larger block sizes.