A box contains M red balls written with numbers from 1 to M, and N green balls written with numbers from 1 to N. A person then draws m red balls and n green balls from this box; later on, after marking down the numbers for each color and then putting the balls back in, a lottery machine also randomly draws m red balls and n green balls. The prize of the lottery counts how many balls there are with the same numbers and colors. Denote the number of matched red balls as i, and the number of matched green balls as j. 1. If we have M = 30, N = 16, m = 6, and n = 1 specifically, what is the probability that one wins the full prize, i.e., P(i = 6 and j = 1)? 2. In general, what is P(i, j)?