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6 problems match “counting subsets”. Clear

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1. 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 $\{4, 5, 6, 7, 8\}$.
2. Expected consecutive pairs in a subset ●●○○○ 20 ptsA 25-element subset is chosen uniformly at random from $\{1,2,\ldots,100\}$. Find the expected number of pairs $(i,i+1)$ of consecutive integers for which both $i$ and $i+1$ lie in the subset.
3. Sizes of two sets from subset counts ●●○○○ 20 ptsLet $A$ and $B$ be two finite sets. The number of subsets of $A$ minus the number of subsets of $B$ is $112$. Find the number of elements in $A$ and in $B$.
4. Spacy subsets of twelve integers ●●●○○ 50 ptsCall a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 12\}$, including the empty set, are spacy?
5. Ten numbered coins sum at least 45 ●●●○○ 50 ptsYou have 10 coins. Coin $i$ shows the number $i$ on one side (for $i = 1, \dots, 10$) and $0$ on the other side. You toss all the coins and add up the values that face up. Find the probability that the sum is greater than or equal to $45$. Give your answer as a decimal.
6. Subsets with smallest element m ●●●●○ 100 ptsDetermine all values of $m$ with $1 \leq m \leq 47$ for which there are exactly $2^{15}$ subsets $X$ of the set $\{1,2,3,\ldots,47\}$ such that $m$ is the smallest element of $X$, and for every $x \in X$, either $x+m \in X$ or $x+m>47$.