KöMaL 2025 C1853

Some researchers are observing a team of fifty bumblebees buzzing on a flowery sandbank. They excitedly note that each bumblebee collected pollen from exactly four types of flowers before flying off. In fact, they also recorded that each bumblebee chose a different set of four flowers, but all $50$ bumblebees visited a common baby's-breath. Prove that the bumblebees collected pollen from at least 9 different types of flowers.