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94 problems match “counting”. Clear

Birthday on a given day ●●○○○ 20 ptsA class has 30 students, and each student's birthday is equally likely to be any of 365 days. Find the probability that at least one student has a birthday on 5 November. Round to three decimal places.
Guaranteed shared birthday count ●●○○○ 20 ptsAmong 10,000 people, what is the largest number of people who are guaranteed to share the same birthday (ignoring the year)? Count February 29 as a possible birthday; the answer is the same with or without it.
Divisors of exactly one number ●●○○○ 20 ptsHow many natural numbers divide exactly one of 1080 and 1800, but not both?
Primes from up to two digits ●●○○○ 20 ptsPick one or two different digits from the set $\{1, 3, 6, 7\}$ and arrange them to form a number. How many prime numbers can be created in this manner?
Composite factors of 420 ●●○○○ 20 ptsFind the number of composite factors of $420$ .
Smallest group with a shared birthday ●●●○○ 50 ptsWhat is the smallest number of people in a group such that the probability that at least two of them share the same birthday is greater than 50%? Assume there are 365 equally likely birthdays and ignore leap years.
Possible resistances from four resistors ●●●○○ 50 ptsRebecca has four resistors, each with resistance 1 ohm. Every minute, she chooses any two resistors with resistances $a$ and $b$ ohms, and combines them into one by one of the following methods: (i) connect them in series, producing a resistor of $a+b$ ohms; (ii) connect them in parallel, producing a resistor of $\frac{ab}{a+b}$ ohms; (iii) short-circuit one of the two resistors, producing a resistor of either $a$ or $b$ ohms. Suppose that after three minutes, Rebecca has a single resistor with resistance $R$ ohms. How many possible values are there for $R$ ?
Almost palindromic letter arrangements ●●●○○ 50 ptsCall a string of letters $S$ an almost palindrome if $S$ and the reverse of $S$ differ in exactly two places. Find the number of ways to order the letters in $HMMTTHEMETEAM$ to get an almost palindrome.
Non-negative elements of a closed set ●●●○○ 50 ptsLet $S$ be the smallest subset of the integers such that $0 \in S$ and, for any $x \in S$ , both $3x \in S$ and $3x+1 \in S$ . Determine the number of non-negative integers in $S$ that are less than $2008$ .
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?