"Happy to submit a resume for you" So What's the Limit?

I had a general informational interview-type call with an alumni. When I talked to him, I didn't have any particular positions in mind, but he said he would be happy to submit a resume for me down the line. I'm trying to make the jump from teaching to consulting (I know pretty much impossible) and so I'm trying to cast a wide net. I see 5 or 6 positions that would interest me in his firm's experienced hire section.

What's a reasonable limit on how many I can ask him to submit? 1? 3? all 6? I plan to ask him in one email to submit them so there's no issue of barraging him with too many emails. Just wondering how many you guys would say is an acceptable amount?

Thanks!