pull-ups

The other day I was doing standing bar curls and getting a super good pump, especially in my inner biceps. A meathead came up to me and said those weren't the best exercises for his arms. He showed me how big his arms were and said, "I didn't get these big arms by doing curls. I got these big...

I was wondering about pull-ups. How many pull-ups can you do? Be honest. The other day my friend challenged me. I was able to do 13 of them and he was able to do 11. I'm talking about pull-ups with good form, not half-assed and cheating.

Can you suggest me a strategy to building up my core. Example I can bench 250 lb no problem but I can't do very many sit-ups or pull-ups. I think that my core is very weak even though I might seem like I'm strong. also do you guys ever train your core?