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Counting & Probability set

20 problems, difficulty 2–4, one per topic where possible

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Counting & Probability setAMC 10 Step by Step

Name: Date: September 13, 2026

20 problems · 60 minutes · 6 / 1.5 / 0

1.2024 AMC 10A #6
What is the minimum number of successive swaps of adjacent letters in the string $ABCDEF$ that are needed to change the string to $FEDCBA?$ (For example, $3$ swaps are required to change $ABC$ to $CBA;$ one such sequence of swaps is $ABC\to BAC\to BCA\to CBA.$ )
1. A)
  $6$
2. B)
  $10$
3. C)
  $12$
4. D)
  $15$
5. E)
  $24$
2.2013 AMC 10A #17
Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next 365-day period will exactly two friends visit her?
1. A)
  $48$
2. B)
  $54$
3. C)
  $60$
4. D)
  $66$
5. E)
  $72$
3.2015 AMC 10B #18
Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads?
1. A)
  $32$
2. B)
  $40$
3. C)
  $48$
4. D)
  $56$
5. E)
  $64$
4.2016 AMC 10A #20
For some particular value of $N$ , when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b,c,$ and $d$ , each to some positive power. What is $N$ ?
1. A)
  $9$
2. B)
  $14$
3. C)
  $16$
4. D)
  $17$
5. E)
  $19$
5.2017 AMC 10A #18
Amelia has a coin that lands heads with probability $\tfrac{1}{3}$ , and Blaine has a coin that lands on heads with probability $\tfrac{2}{5}$ . Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is $\tfrac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. What is $q-p$ ?
1. A)
  $1$
2. B)
  $2$
3. C)
  $3$
4. D)
  $4$
5. E)
  $5$
6.2023 AMC 10B #10
You are playing a game. A $2$ $\times$ $1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3$ $\times$ $3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle?
1. A)
  $3$
2. B)
  $5$
3. C)
  $4$
4. D)
  $8$
5. E)
  $6$
7.2024 AMC 10B #12
A group of $100$ students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students $A$ and $B$ , student $A$ speaks some language that student $B$ does not speak, and student $B$ speaks some language that student $A$ does not speak. What is the least possible total number of languages spoken by all the students?
1. A)
  $9$
2. B)
  $10$
3. C)
  $12$
4. D)
  $51$
5. E)
  $100$
8.2024 AMC 10A #17
Two teams are in a best-two-out-of-three playoff: the teams will play at most $3$ games, and the winner of the playoff is the first team to win $2$ games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a $\frac{2}{3}$ chance of winning at home, and its probability of winning when playing away from home is $p$ . Outcomes of the games are independent. The probability that Team A wins the playoff is $\frac{1}{2}$ . Then $p$ can be written in the form $\frac{1}{2}(m - \sqrt{n})$ , where $m$ and $n$ are positive integers. What is $m+n$ ?
1. A)
  $10$
2. B)
  $11$
3. C)
  $12$
4. D)
  $13$
5. E)
  $14$
9.2025 AMC 10B #16
A circle has been divided into $6$ sectors of different sizes. Then $2$ of the sectors are painted red, $2$ painted green, and $2$ painted blue so that no two neighboring sectors are painted the same color. One such coloring is shown below.

How many different colorings are possible?
1. A)
  $12$
2. B)
  $16$
3. C)
  $18$
4. D)
  $24$
5. E)
  $28$
10.2025 AMC 10A #16
There are three jars. Each of three coins is placed in one of the three jars, chosen at random and independently of the placements of the other coins. What is the expected number of coins in a jar with the most coins?
1. A)
  $\frac{4}{3}$
2. B)
  $\frac{13}{9}$
3. C)
  $\frac{5}{3}$
4. D)
  $\frac{17}{9}$
5. E)
  $2$
11.2010 AMC 10A #22
Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?
1. A)
  $28$
2. B)
  $56$
3. C)
  $70$
4. D)
  $84$
5. E)
  $140$
12.2011 AMC 10A #21
Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine?
1. A)
  $\frac{7}{11}$
2. B)
  $\frac{9}{13}$
3. C)
  $\frac{11}{15}$
4. D)
  $\frac{15}{19}$
5. E)
  $\frac{15}{16}$
13.2013 AMC 10B #17
Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?
1. A)
  $62$
2. B)
  $82$
3. C)
  $83$
4. D)
  $102$
5. E)
  $103$
14.2015 AMC 10A #22
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
1. A)
  $\dfrac{47}{256}$
2. B)
  $\dfrac{3}{16}$
3. C)
  $\dfrac{49}{256}$
4. D)
  $\dfrac{25}{128}$
5. E)
  $\dfrac{51}{256}$
15.2018 AMC 10B #22
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $[0,1]$ . Which of the following numbers is closest to the probability that $x,y,$ and $1$ are the side lengths of an obtuse triangle?
1. A)
  $0.21$
2. B)
  $0.25$
3. C)
  $0.29$
4. D)
  $0.50$
5. E)
  $0.79$
16.2018 AMC 10B #18
Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?
1. A)
  $60$
2. B)
  $72$
3. C)
  $92$
4. D)
  $96$
5. E)
  $120$
17.2018 AMC 10A #20
A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called $\textit{symmetric}$ if its look does not change when the entire square is rotated by a multiple of $90 ^{\circ}$ counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
1. A)
  $\text{ 510}$
2. B)
  $\text{ 1022}$
3. C)
  $\text{ 8190}$
4. D)
  $\text{ 8192}$
5. E)
  $\text{ 65,534}$
18.2019 AMC 10A #20
The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?
1. A)
  $\dfrac{1}{21}$
2. B)
  $\dfrac{1}{14}$
3. C)
  $\dfrac{5}{63}$
4. D)
  $\dfrac{2}{21}$
5. E)
  $\dfrac{1}{7}$
19.2021 AMC 10B #22
Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m + n ?$
1. A)
  $47$
2. B)
  $94$
3. C)
  $227$
4. D)
  $471$
5. E)
  $542$
20.2021 AMC 10B #23
A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square and lands in a random location where the coin is completely contained within the square. The probability that the coin will cover part of the black region of the square can be written as $\frac{1}{196}\left(a+b\sqrt{2}+\pi\right)$ , where $a$ and $b$ are positive integers. What is $a+b$ ?
1. A)
  $64$
2. B)
  $66$
3. C)
  $68$
4. D)
  $70$
5. E)
  $72$

Counting & Probability set — answersAMC 10 Step by Step

Name: Date: September 13, 2026

Keep this page separate from the problems.

Answers

#	Answer	Topic and the idea it turns on
1	D	Games & Processes — Each adjacent swap reverses the order of exactly one pair of letters, and all C(6,2) = 15 pairs must be reversed, so 15 swaps are necessary and sufficient.
2	B	Inclusion-Exclusion — Two specific friends coincide on multiples of their lcm (12, 15 or 20); subtract the multiples of 60, where all three come, from each count.
3	D	Expected Value — A coin stays tails only after three tails in a row, probability 1/8; so each coin is heads with probability 7/8, and 64 * 7/8 = 56.
4	B	Distributions & Stars and Bars — All-four-variables terms are solutions of i+j+k+l+m = N with i,j,k,l >= 1 and m >= 0 for the 1; stars and bars gives C(N,4).
5	D	Conditional Probability & States — If both first tosses fail (probability 2/5), the game restarts from the same state, so p = 1/3 + (2/5) p, giving p = 5/9.
6	C	Paths & Grids — Every domino contains an edge-middle square, so those four squares always work; three squares touch at most 4+3+3 of the 12 dominoes.
7	A	Basic Counting — Students correspond to distinct k-subsets of n languages; the most such subsets is C(n, n/2), and C(8,4) = 70 < 100 <= 126 = C(9,4).
8	E	Basic Probability — Summing the three winning scenarios, (2/3)p + (2/3)(1−p)p + (1/3)p² = 1/2 gives 2p² − 8p + 3 = 0, so p = (4 − √10)/2.
9	D	Arrangements with Restrictions — A colouring is a pairing of the six sectors into three non-neighbouring pairs together with a naming of the colours, and there are exactly 4 such pairings.
10	D	Expected Value — Sort the 27 placements by shape: 3 give 3-0-0, 6 give 1-1-1, 18 give 2-1-0, so the expectation is (9 + 6 + 36)/27.
11	A	Basic Counting — An interior triangle needs three chords that pairwise cross inside the circle, which uses six distinct endpoints, and any six points on the circle produce exactly one such triangle.
12	D	Conditional Probability & States — The pairs weigh the same only when both are all genuine or each contains exactly one counterfeit, so compare just those two counts.
13	E	Games & Processes — Exchanges stop only at 1 red and 2 blue; solving 75-2x+y = 1, 75+x-3y = 2 gives x+y = 103 exchanges.
14	A	Recursive Counting — Count circular strings with no two adjacent 1s: fix person 1; seated gives a line of 7 (34 ways), standing gives a line of 5 (13 ways), total 47.
15	C	Geometric Probability — With 1 the longest side, the conditions are x + y > 1 and x^2 + y^2 < 1: a quarter disk minus a triangle, area pi/4 - 1/2.
16	D	Arrangements with Restrictions — Siblings sharing a row forces a front-back sibling pair, so each row has one child per family: 2^3 * 3! front rows, then 2 derangements behind.
17	B	Paths & Grids — Full square symmetry means the code is determined by one-eighth of the grid: a triangular wedge of 10 cells, so 2^10 colorings minus the two monochrome ones.
18	B	Basic Probability — Only parity matters: the five odd numbers must fill one full row and one full column (a cross), 9 placements out of C(9,5) = 126.
19	D	Inclusion-Exclusion — Fix Ang's placement; box i is monochromatic iff Ben's and Jasmin's permutations both fix i, so count pairs with a common fixed point by inclusion–exclusion.
20	C	Geometric Probability — The center lives in a 7×7 square; it hits black iff within 1/2 of a black shape, so expand the diamond by 1/2 and clip each corner triangle.