Binary fraction and orbital characteristics provide indications on the conditions of star formation, as they shed light on the environment they were born in. Multiple systems are more common in low density environments than in higher density environments. In the current debate about the formation of globular clusters and their multiple populations, studying the binary incidence in the populations they host offers a crucial piece of information on the environment of their birth and their subsequent dynamical evolution. Through a multiyear observational campaign using FLAMES at VLT, we monitored the radial velocity of 968 red-giant-branch stars located around the half-light radii in a sample of ten Galactic globular clusters. We found a total of 21 radial velocity variables identified as bona fide binary stars, for a binary fraction of 2.2% ± 0.5%. When separating the sample into first generation and second generation stars, we find a binary fraction of 4.9% ± 1.3% and 1.2% ± 0.4%, respectively. Through simulations that take possible sources of bias into account in detecting radial velocity variations in the two populations, we show that the difference is significant and only marginally affected by these effects. This kind of different binary fraction strongly suggests different conditions in the environment of formation and evolution of first and second generations stars, with the latter being born in a much denser environment. Our result hence strongly supports the idea that the second generation forms in a dense subsystem at the center of the loosely distributed first generation, where (loose) binaries are efficiently destroyed.