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Stage 3 · Combinatorics

10 problems · AMC 10/12, early questions · mathsolympiadprep.com

Print or save as PDF The answer key prints on its own page at the end.

1. There are 6 identical cards, with the numbers "$1921$", "$1994$", "$1935$", "$1949$", "$1978$", and "$1980$" written on them. These cards are placed with their backs facing up, shuffled, and one card is randomly drawn. The probability of drawing a card with an even number on it is ____.

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2. For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$, and $C$ are sets for which $n(A)+n(B)+n(C)=n(A\cup B\cup C)$ and $|A|=|B|=100$, then what is the minimum possible value of $|A\cap B\cap C|$?

   1. A96
   2. B97
   3. C98
   4. D99
   5. E100

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3. An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is \$20$ or more?
   $

   1. A${{{\frac{1}{4}}}}$
   2. B${{{\frac{2}{5}}}}$
   3. C${{{\frac{3}{7}}}}$
   4. D${{{\frac{1}{2}}}}$
   5. E${{{\frac{2}{3}}}}$

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4. Given the universal set $U={1,2,3,4}$, set $A={1,2,3}$, and set $B={2,3,4}$, find the complement of $A \cap B$ in $U$, denoted as $(A \cap B)^c_U =$ _______ .

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5. During the epidemic, a student took online courses through a web platform and studied at home. One morning, he had four online classes scheduled, which were mathematics, Chinese, politics, and geography. In the afternoon, he had three classes scheduled, which were English, history, and physical education. Now, he is planning to check in for one class in the morning and one class in the afternoon. The probability that at least one of the two selected classes is a subject related to arts and sciences (politics, history, geography) is:

   1. A$\frac{3}{4}$
   2. B$\frac{7}{12}$
   3. C$\frac{2}{3}$
   4. D$\frac{5}{6}$

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6. An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?

   1. A120
   2. B121
   3. C221
   4. D230
   5. E231

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7. For sets $A$ and $B$, let $card(A)$ represent the number of elements in the finite set $A$. Given $card(A) = M$ and $card(B) = N$ $(M < N)$, and set $C$ satisfies $A \subseteq C \subseteq B$, the number of sets $C$ that meet the conditions is ______.

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8. Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game?

   1. A$\frac{1}{5}$
   2. B$\frac{1}{4}$
   3. C$\frac{1}{3}$
   4. D$\frac{1}{2}$
   5. E$\frac{2}{3}$

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9. Diana and Apollo each roll a standard die obtaining a number at random from $1$ to $6$. What is the probability that Diana's number is larger than Apollo's number?

   1. A$\dfrac{1}{3}$
   2. B$\dfrac{5}{12}$
   3. C$\dfrac{4}{9}$
   4. D$\dfrac{17}{36}$
   5. E$\dfrac{1}{2}$

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10. A cricket randomly hops between $4$ leaves, on each turn hopping to one of the other $3$ leaves with equal probability. After $4$ hops what is the probability that the cricket has returned to the leaf where it started?

    2022 AMC 8 Problem 25 Picture.jpg

    1. A$\frac{2}{9}$
    2. B$\frac{19}{80}$
    3. C$\frac{20}{81}$
    4. D$\frac{1}{4}$
    5. E$\frac{7}{27}$

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Answer key — Stage 3 · Combinatorics

Worked solutions for every problem are on the site, one page per problem.

1. $\frac{1}{2}$ open
2. $97$ open
3. $\frac{1}{2}$ open
4. ${1,4}$ open
5. $C$ open
6. $121$ open
7. $2^{N-M}$ open
8. $\frac{1}{5}$ open
9. $\frac{5}{12}$ open
10. $\frac{7}{27}$ open

Problems belong to the competitions that set them and are reproduced from open datasets under their licences; every problem page names its source. Free to copy for classroom use.