He didn't. God was in charge of selecting the animals Noah was there to make sure that there were two unclean animals and seven clean ones.
It was not primarily Noah's project, it was Jehovah God's - HE gave the specifications of the ark, so many cubits, here and there, and he even said what kind of wood to use, and to use tar to cover it all.
Noah was the obedient one, and his following his God's instructions "just so" made it possible for humanity to continue living. Plus, God saw to it that the animals that HE wanted to save went into the ark.
God is the creator of heredity, and DNA. He could make all birds from a representative few, all cats from a few, all horses, elephants, and so on.
Researchers estimate that there are only about 290 species of land mammals larger than sheep and about 1,360 smaller than rats. (The Deluge Story in Stone, by B. C. Nelson, 1949, p. 156; The Flood in the Light of the Bible, Geology, and Archaeology, by A. M. Rehwinkel, 1957, p. 69) So, even if estimates are based on these expanded figures, the ark could easily have accommodated a pair of all these animals.