A lottery math problem

The MegaMillion Lotto Jackpot is now $237 Million. The odds of winning are about 1:175 Million. This means that if Rocky fills out 175 million lottery tickets, he is guaranteed to make a profit. (Assuming that he doesn’t have to share the prize.)

http://www.nylottery.org/ny/nyStore/cgi-bin/ProdSubEV_Cat_403_SubCat_337550_NavRoot_320.htm
http://www.megamillions.com/

But Rocky doesn’t want to stand at his local Seven-Eleven and fill out 175 Million tickets. (After he enters  Trophy Wife’s birthdate, the dog’s birthdate,  and his lucky number from inside of a Chinese Fortune Cookie, he won’t remember what numbers to pick.)  So instead, he will ask the Seven-Eleven lottery clerk for  175 Million “Quick Pick” tickets.

A “Quick-Pick” is a computer-generated random number lottery entry. The computer picks the numbers, so Rocky doesn’t have to think that hard.

Alas, this won’t work either. Because even if Rocky buys 175 million Quick-Pick, there is some chance that he will receive duplicate Quick-Pick entries … and there is some chance that he won’t receive the winning combination.

The chance of getting a duplicate Quick-Pick should be the same as the chance of winning the lottery. But in Rocky’s case, achieving this result is an illustration of really bad luck.

So Rocky poses the following math question: What is the OPTIMAL number of Quick-Pick tickets to buy? (The optimal number should maximize the chance of getting the winning combination, and minimize the chance of getting a duplicate combination.)

As always, the reader with the best submission will receive a unique prize of dubious monetary value.