What is duality, and what is the best example of it?

Can you help me on letters E-H

Roll a fair sided die three times. Let A1 = {1 or 2 on the first roll }, A2 = {3 or 4 on the second roll}, and A3 = {5 or 6 on the third roll}. It is given that P(Ai ) = 1/3, i = 1,2,3; P(Ai intersection Aj) = (1/3)2 , i cannot equal j; and P(A1 intersection A2 intersection A3) = (1/3)3 a) use theorem 1.1-6 to find P(A1 union A2 union A3) b) show that P(A1 union A2 union A3) = 1-(1-1/3)3

Sherri's Cherries makes both white and dark chocolate cherries for Christmas. Each white cherry uses one cherry and 2 grams of white chocolate, while each dark cherry uses one cherry and 3 grams of dark chocolate. Profit is 7¢ per white cherry and 8¢ per dark cherry. Each hour Sherri has 100 cherries, 270 grams of dark chocolate, and 160 grams of white chocolate. How many cherries of each type should be made per hour to maximize profit? a) 80 white, 90 dark b) 10 white, 90 dark c) O white, 100 dark d) 80 white, 20 dark e) None of the others

(Devore Section 2.3 #30) A friend of mine is giving a dinner party. His current wine supply includes 8 bottles of zinfandel, 10 of merlot, and 12 of cabernet (he only drinks red wine), all from different wineries. (a) If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this? (b) If 6 bottles of wine are to be randomly selected from the 30 for serving, how many ways are there to do this? (c) If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety? (d) If 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen? (e) If 6 bottles are randomly selected, what is the probability that all of them are the same variety?