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11 problems match “inclusion-exclusion”. Clear

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1. Time with exactly two candles burning ●●○○○ 20 ptsThree candles can burn for $30$, $40$, and $50$ minutes respectively, but they are not lit at the same time. Each candle burns until it is used up. The three candles are burning simultaneously for a total of $10$ minutes, and exactly one candle is burning for a total of $20$ minutes. Find the total time, in minutes, during which exactly two candles are burning simultaneously.
2. Subsets contained in neither of two sets ●●○○○ 20 ptsFind the number of subsets of $\{1, 2, 3, 4, 5, 6, 7, 8\}$ that are subsets of neither $\{1, 2, 3, 4, 5\}$ nor $\{4, 5, 6, 7, 8\}$.
3. Divisors of exactly one number ●●○○○ 20 ptsHow many natural numbers divide exactly one of 1080 and 1800, but not both?
4. Every month has a birthday ●●●○○ 50 ptsAssume 40 people are independent and each birthday is equally likely to fall in any of the 12 months. What is the probability that every one of the 12 months contains at least one of the 40 birthdays? Give your answer as a decimal rounded to seven decimal places.
5. Partition of a set with size conditions ●●●○○ 50 ptsLet $N$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ such that $\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}$, $\mathcal{A} \cap \mathcal{B} = \emptyset$, the number of elements of $\mathcal{A}$ is not an element of $\mathcal{A}$, and the number of elements of $\mathcal{B}$ is not an element of $\mathcal{B}$. Find $N$.
6. Arranging letters of COMPUTE ●●●○○ 50 ptsHow many ways are there to arrange the letters of the word COMPUTE so that the vowels are in alphabetical order, or the consonants are in alphabetical order, or at least two vowels are next to each other?
7. Exactly two friends visiting Daphne ●●●○○ 50 ptsDaphne is visited periodically by her three best friends, Alice, Beatrix and Claire. Alice visits every third day, Beatrix every fourth day, and Claire every fifth day. All three friends visited Daphne yesterday. On how many days of the next 365-day period will exactly two friends visit her?
8. Six-letter words using all letters ●●●○○ 50 ptsSix-letter "words" are formed using the letters A, B, C and D, with letters allowed to repeat. In how many of them does each letter appear at least once?
9. Counting super keys from dependencies ●●●○○ 50 ptsA relation $R$ has attributes $A, B, C, D, E, F, G, H$ and the functional dependencies $CH \to G$, $A \to BC$, $B \to CFH$, $E \to A$, $F \to EG$. How many super keys does $R$ have? A super key is a set of attributes whose closure contains all attributes of $R$.
10. Smallest p for triple intersection ●●●○○ 50 ptsLet events $A$, $B$ and $C$ satisfy $P(A)=P(B)=P(C)=p$. Find the infimum of the values of $p$ for which $P(A\cap B\cap C)>0.95$ holds for every possible joint distribution of $A$, $B$ and $C$. Round to four decimal places.

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