4.4 #6. Ace or Heart - that's 4 + 13 - 1 possibilities. So (4+13-1)/52 = 4/13.
4.4 #12. Exactly one ace. First pick the ace (4), then pick the other cards (52-aces choose 4). So 4 * (48 choose 4) / (52 choose 5).
4.4 #28. Of the (80 choose 7) lottery ticket possibilities, only (11 choose 7) win. So (11 choose 7) / (80 choose 7).
4.4 #32. Rolling an 8 on two dice: 2+6, 3+5, 4+4, 5+3, 6+2, so 5 out of 6^2. Rolling an 8 on three dice: 6+1+1, 5+2+1, 5+1+2, 4+(three ways), 3+(four ways), 2+(five ways), 1+(six ways), so 21/6^3. Then 5/6^2 is about 20% but 21/6^3 is about 10%. So two dice.
4.6 #8. 12 stars, 20 bars, so (32 choose 12).
4.6 #20. With the x_4, it's 11 stars, three bars, so (14 choose 3).
4.6 #28. If the AAA are consecutive, it's silly to think of them as separate letters. Really, we're just combining AAA, R, R, D, V, K. So it's the multinomial coefficient (6 choose 1,2,1,1,1) = 6! / 2!. If you want, you can briefly replace the second R with a backwards R (as in Aardvark's 'R' Us), and then you're just arranging 6 different things, so 6!. Then you forget the difference, which divides by 2.