One A trades for 10 B, one B trades for 10 C, but somehow one A trades for only 50 C when exchanged directly.
There's a clear arbitrage opportunity here: Sell one A for 10 B, sell the 10 B for 100 C, and finally go to the person willing to give you an A for 50 C – and buy two. You started out with one A and ended up with two. This is great! Now repeat that until the arbitrage opportunity disappears and get rich in the process.
Now here's the question/idea: If you can do these three trades atomically, why do you need a single A to being with?
This is what "flash loans" (or any equivalent system involving multilateral netting) allow.