Why can an n-p-n transistor send many electrons to the collector for each electron reaching the base?

In an n-p-n transistor, the ability to send many electrons to the collector for each electron reaching the base is a result of the device's specific structural design and biasing configuration.12

The transistor is constructed with two closely coupled p-n junctions. In normal operation, the emitter-base junction is forward-biased, causing electrons to be injected from the emitter into the narrow p-type base region. Because the base is very thin, most of these injected electrons do not recombine with the holes in the base. Instead, they are swept across the base-collector junction by the strong electric field created by the reverse bias applied to that junction.21

Because the collector is strongly reverse-biased, it effectively "collects" the vast majority of the electrons injected from the emitter. Consequently, only a small fraction of the current flows out through the base connection, while a much larger current flows to the collector, resulting in the current amplification characteristic of the common-emitter configuration.12