Cancer incidence increase has multiple aetiologies. Mutant alleles accumulation in populations may be one of them due to strong heritability of many cancers. The opportunity for the operation of natural selection has decreased in the past ~150 years because of reduction in mortality and fertility. Mutation-selection balance may have been disturbed in this process and genes providing background for some cancers may have been accumulating in human gene pools. Worldwide, based on the WHO statistics for 173 countries the index of the opportunity for selection is strongly inversely correlated with cancer incidence in peoples aged 0–49 years and in people of all ages. This relationship remains significant when gross domestic product per capita (GDP), life expectancy of older people (e50), obesity, physical inactivity, smoking and urbanization are kept statistically constant for fifteen (15) of twenty-seven (27) individual cancers incidence rates. Twelve (12) cancers which are not correlated with relaxed natural selection after considering the six potential confounders are largely attributable to external causes like viruses and toxins. Ratios of the average cancer incidence rates of the 10 countries with lowest opportunities for selection to the average cancer incidence rates of the 10 countries with highest opportunities for selection are 2.3 (all cancers at all ages), 2.4 (all cancers in 0–49 years age group), 5.7 (average ratios of strongly genetically based cancers) and 2.1 (average ratios of cancers with less genetic background).