I have attempted to solve these questions, but can someone double check my answers please?
Five identical dice are rolled simultaneously.
a) How many different outcomes are possible? C(10,5) = 252
b) How many outcomes have three or more dice showing the same value? 6 + 6 * 5 + 6 * 5^2 = 186
c) How many outcomes are a "full house" (i.e., three dice have the same value, and the other two dice have the same value, but not all five dice have the same value)? 6 * 5 = 30a) How many ways can the letters in HALLOWEEN be arranged? P(9; 2, 2, 1, 1, 1, 1, 1) or 90,720
b) What if we require the vowels to be in alphabetical order (but not necessarily in consecutive positions)? Example for (b): HALELEOWN and HAEEOLNWL are allowed, LAOEEHLWN and HOELAELWN are not. C(9, 4) * P(5; 2, 1, 1, 1) ways or 7560 ways.
I agree with everything except b). Let's suppose the repeating number is 1(I can say "the" repeating number because two different numbers cannot repeat thrice or more, since the dices are 5): $$1,1,1,\square,\square$$ There are $21$ ways to fill the squares since the couples $(a,a), a\in\{0,1,2,3,4,5,6\}$ are $6$ and the couples $(a,b); a\neq b;a,b\in\{0,1,2,3,4,5,6\}$ are $6 \times 5=30$. So: $$6+30/2=21$$ The procedure can be repeated for any repeating number, so the answer should be $6\times 21=126$