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Basic Counting

Multiplication principle, permutations, combinations, counting by listing

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Basic CountingAMC 10 Step by Step

Name: Date: October 28, 2026

all 57 problems, easiest first

What you need to know

Multiplication principle. If a job has stages, and each stage has the same number of choices no matter what you chose before, multiply. Cases that do not overlap add. Example: $2$ letters then $3$ digits make $26\cdot26\cdot10\cdot10\cdot10 = 676{,}000$ license plates.
Permutations (order matters). $n$ different objects can be put in order in $n!$ ways. Picking $k$ of them in order: $P(n,k) = \frac{n!}{(n-k)!}$ ways. Example: a president, vice president, and secretary from $10$ people: $10\cdot9\cdot8 = 720$ .
Combinations (order does not matter). Choosing $k$ of $n$ different objects: $\binom{n}{k} = \frac{n!}{k!\,(n-k)!}$ . Also $\binom{n}{k} = \binom{n}{n-k}$ (choosing who is in is choosing who is out), and $\binom n0+\binom n1+\dots+\binom nn = 2^n$ (each object is in or out). Example: a $3$ -person committee from $10$ people: $\frac{720}{3!} = 120$ , since each committee has $3!$ orders.
Repeated objects: $\frac{n!}{a!\,b!\cdots}$ , where $a$ , $b$ , … count each group of identical objects. Example: ALABAMA has four A's, so $\frac{7!}{4!} = 210$ arrangements.
Complementary counting. Count what you do not want and subtract from the total. Example: $5^4 = 625$ four-digit numbers use only odd digits, so $9000-625 = 8375$ have an even digit.

How AMC 10 tests it

Problems 3–10: "How many three-digit numbers / license plates / subsets have property X?", by multiplying or by one $\binom{n}{k}$ .
Problems 10–18: choose-then-arrange questions (committees, handshakes, seating) where the trap is whether order matters.
Digit problems that come down to short casework on the first digit.
Counting the right thing: pairs of people, diagonals ( $\binom{n}{2}-n$ : pairs of corners minus the sides), crossing points of chords ( $\binom{n}{4}$ for $n$ points on a circle, if no three chords meet at one point).
The count is often the hidden step inside a probability problem.

Standard approaches

Decide whether order matters. Then build the object stage by stage and multiply.
If the total is quick to find, count the opposite and subtract.
Split into cases that do not overlap, based on the most limited spot (first digit, largest number, a special object), and add.
Look for a one-to-one correspondence with something you can count directly: strings of $0$ s and $1$ s, subsets, grid paths. Example: subsets of $\{1,2,3,4,5\}$ match the $0$ - $1$ strings of length $5$ ( $1$ means "in"), so there are $2^5 = 32$ .
Check with a tiny case you can list by hand.

Worked example

How many four-digit positive integers have digits that are strictly increasing from left to right or strictly decreasing from left to right?

(A) $126$ (B) $210$ (C) $330$ (D) $336$ (E) $420$

Solution. Any four different digits can be written in increasing order in exactly one way, and in decreasing order in one way. So we count sets of four digits.

Step 1: increasing. A $0$ would be the smallest digit, so it would come first, which is not allowed. Choose $4$ digits from $1$ to $9$ : $\binom94 = 126$ .

Step 2: decreasing. Now a $0$ would come last, which is fine. Choose $4$ digits from $0$ to $9$ : $\binom{10}{4} = 210$ . (For example, $\{0,3,5,9\}$ gives $9530$ .)

Step 3: add. No number is both: $126+210 = 336$ .

The answer is $\boxed{\textbf{(D)}\ 336}$ .

Pitfalls

Multiplying when one stage's choices depend on an earlier choice. Use casework. Example: in a two-digit number whose second digit is bigger than the first, the second digit has $8$ choices after a $1$ but $1$ choice after an $8$ .
Mixing up $\binom{n}{k}$ and $P(n,k)$ . A committee has no order; a line-up does.
Forgetting that the first digit cannot be $0$ , while $0$ is fine elsewhere. There are $900$ three-digit numbers, not $1000$ .
Counting things twice in "at least one" problems. Example: $3$ -person committees from $5$ boys and $5$ girls with at least one girl: "pick a girl, then any $2$ others" gives $5\cdot\binom92 = 180$ , but the answer is $\binom{10}{3}-\binom53 = 110$ . Complementary counting avoids this.

Traps that recur

Summing binom(9,k) 2^k instead of binom(9,k) 2^(k-1), which gives 3^9 - 1 = 19682, choice (E); the factor 2^(k-1) is right because the largest digit's position is forced. (2025 AMC 10A #24)
Multiplying the 81 multiples of 11 by the number of permutations, which double counts triples like {2, 0, 9} that give two different multiples (209 and 902). (2017 AMC 10A #25)
Answering 70 by assuming no three diagonals are concurrent; the regular octagon has concurrency at the center and at eight other points. (2013 AMC 10A #25)
Forgetting that the two-friends case includes two separate triangles as well as one six-person loop, or overcounting loops by ignoring rotations and reflections. (2012 AMC 10A #23)

1.2024 AMC 10B #1
In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?
1. A) $2021$
2. B) $2022$
3. C) $2023$
4. D) $2024$
5. E) $2025$
2.2018 AMC 10B #3
In the expression $\left(\underline{\qquad}\times\underline{\qquad}\right)+\left(\underline{\qquad}\times\underline{\qquad}\right)$ each blank is to be filled in with one of the digits $1,2,3,$ or $4,$ with each digit being used once. How many different values can be obtained?
1. A) $2$
2. B) $3$
3. C) $4$
4. D) $6$
5. E) $24$
3.2018 AMC 10B #1
Kate bakes a $20$ -inch by $18$ -inch pan of cornbread. The cornbread is cut into pieces that measure $2$ inches by $2$ inches. How many pieces of cornbread does the pan contain?
1. A) $90$
2. B) $100$
3. C) $180$
4. D) $200$
5. E) $360$
4.2018 AMC 10B #5
How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?
1. A) $128$
2. B) $192$
3. C) $224$
4. D) $240$
5. E) $256$
5.2008 AMC 10B #1
A basketball player made $5$ baskets during a game. Each basket was worth either $2$ or $3$ points. How many different numbers could represent the total points scored by the player?
1. A) $2$
2. B) $3$
3. C) $4$
4. D) $5$
5. E) $6$
6.2004 AMC 10B #1
Each row of the Misty Moon Amphitheater has $33$ seats. Rows $12$ through $22$ are reserved for a youth club. How many seats are reserved for this club?
1. A) $297$
2. B) $330$
3. C) $363$
4. D) $396$
5. E) $726$
7.2004 AMC 10B #2
How many two-digit positive integers have at least one $7$ as a digit?
1. A) $10$
2. B) $18$
3. C) $19$
4. D) $20$
5. E) $30$
8.2024 AMC 10A #9
In how many ways can $6$ juniors and $6$ seniors form $3$ disjoint teams of $4$ people so that each team has $2$ juniors and $2$ seniors?
1. A) $720$
2. B) $1350$
3. C) $2700$
4. D) $3280$
5. E) $8100$
9.2023 AMC 10A #6
An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is $21$ . What is the value of the cube?
1. A) $42$
2. B) $63$
3. C) $84$
4. D) $126$
5. E) $252$
10.2022 AMC 10B #3
How many three-digit positive integers have an odd number of even digits?
1. A) $150$
2. B) $250$
3. C) $350$
4. D) $450$
5. E) $550$
11.2020 AMC 10A #6
How many $4$ -digit positive integers (that is, integers between $1000$ and $9999$ , inclusive) having only even digits are divisible by $5?$
1. A) $80$
2. B) $100$
3. C) $125$
4. D) $200$
5. E) $500$
12.2020 AMC 10B #5
How many distinguishable arrangements are there of $1$ brown tile, $1$ purple tile, $2$ green tiles, and $3$ yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.)
1. A) $210$
2. B) $420$
3. C) $630$
4. D) $840$
5. E) $1050$
13.2017 AMC 10A #8
At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?
1. A) $240$
2. B) $245$
3. C) $290$
4. D) $480$
5. E) $490$
14.2015 AMC 10A #3
Ann made a 3-step staircase using 18 toothpicks as shown in the figure. How many toothpicks does she need to add to complete a 5-step staircase?
1. A) $9$
2. B) $18$
3. C) $20$
4. D) $22$
5. E) $24$
15.2013 AMC 10A #11
A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly 10 ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person planning committee be selected?
1. A) $10$
2. B) $12$
3. C) $15$
4. D) $18$
5. E) $25$
16.2013 AMC 10A #7
A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?
1. A) $6$
2. B) $8$
3. C) $9$
4. D) $12$
5. E) $16$
17.2011 AMC 10A #13
How many even integers are there between 200 and 700 whose digits are all different and come from the set {1, 2, 5, 7, 8, 9}?
1. A) $12$
2. B) $20$
3. C) $72$
4. D) $120$
5. E) $200$
18.2009 AMC 10B #11
How many $7$ -digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$ , $2$ , $3$ , $3$ , $5$ , $5$ , $5$ ?
1. A) $6$
2. B) $12$
3. C) $24$
4. D) $36$
5. E) $48$
19.2009 AMC 10B #7
By inserting parentheses, it is possible to give the expression $2\times3 + 4\times5$ several values. How many different values can be obtained?
1. A) $2$
2. B) $3$
3. C) $4$
4. D) $5$
5. E) $6$
20.2007 AMC 10A #12
Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?
1. A) $56$
2. B) $58$
3. C) $60$
4. D) $62$
5. E) $64$
21.2007 AMC 10B #8
On the trip home from the meeting where this AMC10 was constructed, the Contest Chair noted that his airport parking receipt had digits of the form $bbcac,$ where $0 \le a < b < c \le 9,$ and $b$ was the average of $a$ and $c.$ How many different five-digit numbers satisfy all these properties?
1. A) $12$
2. B) $16$
3. C) $18$
4. D) $20$
5. E) $25$
22.2004 AMC 10A #12
Henry's Hamburger Haven offers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two,or three meat patties and any collection of condiments. How many different kinds of hamburgers can be ordered?
1. A) $24$
2. B) $256$
3. C) $768$
4. D) $40,320$
5. E) $120,960$
23.2004 AMC 10A #13
At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party?
1. A) $8$
2. B) $12$
3. C) $16$
4. D) $18$
5. E) $24$
24.2003 AMC 10B #10
Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times has the number of possible license plates increased?
1. A) $\frac{26}{10}$
2. B) $\frac{26^2}{10^2}$
3. C) $\frac{26^2}{10}$
4. D) $\frac{26^3}{10^3}$
5. E) $\frac{26^3}{10^2}$
25.2003 AMC 10B #16
A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that a restaurant should offer so that a customer could have a different dinner each night in the year $2003$ ?
1. A) $4$
2. B) $5$
3. C) $6$
4. D) $7$
5. E) $8$
26.2002 AMC 10B #9
Using the letters $A$ , $M$ , $O$ , $S$ , and $U$ , we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word" $USAMO$ occupies position
1. A) $112$
2. B) $113$
3. C) $114$
4. D) $115$
5. E) $116$
27.2025 AMC 10B #9
How many ordered triples of integers $(x, y, z)$ satisfy the following system of inequalities?
$\begin{align*} -x-y-z&\le -2\\ -x+y+z&\le 2\\ x-y+z&\le 2\\ x+y-z&\le 2 \end{align*}$
1. A) $4$
2. B) $8$
3. C) $11$
4. D) $15$
5. E) $17$
28.2025 AMC 10A #12
Carlos uses a $4$ -digit passcode to unlock his computer. In his passcode, exactly one digit is even, exactly one (possibly different) digit is prime, and no digit is $0$ . How many $4$ -digit passcodes satisfy these conditions?
1. A) $176$
2. B) $192$
3. C) $432$
4. D) $464$
5. E) $608$
29.2024 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) $100$
30.2023 AMC 10B #16
Define an $upno$ to be a positive integer of $2$ or more digits where the digits are strictly increasing moving left to right. Similarly, define a $downno$ to be a positive integer of $2$ or more digits where the digits are strictly decreasing moving left to right. For instance, the number $258$ is an upno and $8620$ is a downno. Let $U$ equal the total number of $upnos$ and let $D$ equal the total number of $downnos$ . What is $|U-D|$ ?
1. A) $512$
2. B) $10$
3. C) $0$
4. D) $9$
5. E) $511$
31.2021 AMC Fall 10A #21
Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\frac{p}{q}$ ?
1. A) $1$
2. B) $4$
3. C) $8$
4. D) $12$
5. E) $16$
32.2021 AMC 10A #15
Values for $A,B,C,$ and $D$ are to be selected from $\{1, 2, 3, 4, 5, 6\}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter; for example, the choices $A=3, B=2, C=4, D=1$ is considered the same as the choices $A=4, B=1, C=3, D=2.$ )
1. A) $30$
2. B) $60$
3. C) $90$
4. D) $180$
5. E) $360$
33.2019 AMC 10A #17
A child builds towers using identically shaped cubes of different colors. How many different towers with a height $8$ cubes can the child build with $2$ red cubes, $3$ blue cubes, and $4$ green cubes? (One cube will be left out.)
1. A) $24$
2. B) $288$
3. C) $312$
4. D) $1,260$
5. E) $40,320$
34.2017 AMC 10B #17
Call a positive integer $\textbf{monotonous}$ if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$ , $23578$ , and $987620$ are monotonous, but $88$ , $7434$ , and $23557$ are not. How many monotonous positive integers are there?
1. A) $1024$
2. B) $1524$
3. C) $1533$
4. D) $1536$
5. E) $2048$
35.2015 AMC 10A #13
Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Claudia have?
1. A) $3$
2. B) $4$
3. C) $5$
4. D) $6$
5. E) $7$
36.2009 AMC 10B #19
A particular $12$ -hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$ , it mistakenly displays a $9$ . For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?
1. A) $\frac 12$
2. B) $\frac 58$
3. C) $\frac 34$
4. D) $\frac 56$
5. E) $\frac 9{10}$
37.2007 AMC 10B #20
A set of $25$ square blocks is arranged into a $5 \times 5$ square. How many different combinations of $3$ blocks can be selected from that set so that no two are in the same row or column?
1. A) $100$
2. B) $125$
3. C) $600$
4. D) $2300$
5. E) $3600$
38.2006 AMC 10A #18
A license plate in a certain state consists of $4$ digits, not necessarily distinct, and $2$ letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible?
1. A) $10^4\times 26^2$
2. B) $10^3\times 26^3$
3. C) $5\times 10^4\times 26^2$
4. D) $10^2\times 26^4$
5. E) $5\times 10^3\times 26^3$
39.2006 AMC 10A #21
How many four-digit positive integers have at least one digit that is a $2$ or a $3$ ?
1. A) $2439$
2. B) $4096$
3. C) $4903$
4. D) $4904$
5. E) $5416$
40.2005 AMC 10B #18
All of David's telephone numbers have the form $555-abc-defg$ , where $a$ , $b$ , $c$ , $d$ , $e$ , $f$ , and $g$ are distinct digits and in increasing order, and none is either $0$ or $1$ . How many different telephone numbers can David have?
1. A) $1$
2. B) $2$
3. C) $7$
4. D) $8$
5. E) $9$
41.2005 AMC 10B #20
What is the average (mean) of all 5-digit numbers that can be formed by using each of the digits 1, 3, 5, 7, and 8 exactly once?
1. A) $48000$
2. B) $49999.5$
3. C) $53332.8$
4. D) $55555$
5. E) $56432.8$
42.2004 AMC 10A #16
The $5\times 5$ grid shown contains a collection of squares with sizes from $1\times 1$ to $5\times 5$ . How many of these squares contain the black center square?
1. A) $12$
2. B) $15$
3. C) $17$
4. D) $19$
5. E) $20$
43.2024 AMC 10B #22
A group of $16$ people will be partitioned into $4$ indistinguishable $4$ -person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r}M$ , where $r$ and $M$ are positive integers and $M$ is not divisible by $3$ . What is $r$ ?
1. A) $5$
2. B) $6$
3. C) $7$
4. D) $8$
5. E) $9$
44.2022 AMC 10A #22
Suppose that $13$ cards numbered $1, 2, 3, \ldots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards $1, 2, 3$ are picked up on the first pass, $4$ and $5$ on the second pass, $6$ on the third pass, $7, 8, 9, 10$ on the fourth pass, and $11, 12, 13$ on the fifth pass. For how many of the $13!$ possible orderings of the cards will the $13$ cards be picked up in exactly two passes?
1. A) $4082$
2. B) $4095$
3. C) $4096$
4. D) $8178$
5. E) $8191$
45.2021 AMC Fall 10B #21
Regular polygons with $5,6,7,$ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?
1. A) $52$
2. B) $56$
3. C) $60$
4. D) $64$
5. E) $68$
46.2020 AMC 10A #19
As shown in the figure below, a regular dodecahedron (the polyhedron consisting of $12$ congruent regular pentagonal faces) floats in empty space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?
1. A) $125$
2. B) $250$
3. C) $405$
4. D) $640$
5. E) $810$
47.2020 AMC 10B #17
There are $10$ people standing equally spaced around a circle. Each person knows exactly $3$ of the other $9$ people: the $2$ people standing next to him or her, as well as the person directly across the circle. How many ways are there for the $10$ people to split up into $5$ pairs so that the members of each pair know each other?
1. A) $11$
2. B) $12$
3. C) $13$
4. D) $14$
5. E) $15$
48.2020 AMC 10B #23
Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations:
$\quad\bullet\qquad$ $L,$ a rotation of $90^{\circ}$ counterclockwise around the origin;
$\quad\bullet\qquad$ $R,$ a rotation of $90^{\circ}$ clockwise around the origin;
$\quad\bullet\qquad$ $H,$ a reflection across the $x$ -axis; and
$\quad\bullet\qquad$ $V,$ a reflection across the $y$ -axis.
Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.)
1. A) $2^{37}$
2. B) $3\cdot 2^{36}$
3. C) $2^{38}$
4. D) $3\cdot 2^{37}$
5. E) $2^{39}$
49.2017 AMC 10A #23
How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$ , inclusive?
1. A) $2128$
2. B) $2148$
3. C) $2160$
4. D) $2200$
5. E) $2300$
50.2012 AMC 10B #24
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?
1. A) $108$
2. B) $132$
3. C) $671$
4. D) $846$
5. E) $1105$
51.2010 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) $140$
52.2008 AMC 10A #23
Two subsets of the set $S=\lbrace a,b,c,d,e\rbrace$ are to be chosen so that their union is $S$ and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter?
1. A) $20$
2. B) $40$
3. C) $60$
4. D) $160$
5. E) $320$
53.2003 AMC 10A #23
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure, we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $2003$ small equilateral triangles?
1. A) $1,004,004$
2. B) $1,005,006$
3. C) $1,507,509$
4. D) $3,015,018$
5. E) $6,021,018$
54.2025 AMC 10A #24
Call a positive integer fair if no digit is used more than once, it has no 0s, and no digit is adjacent to two greater digits. For example, $196$ , $23$ , and $12463$ are fair, but $1546$ , $320$ , and $34321$ are not fair. How many fair positive integers are there?
1. A) $511$
2. B) $2584$
3. C) $9841$
4. D) $17711$
5. E) $19682$
55.2017 AMC 10A #25
How many integers between $100$ and $999$ , inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999?$ For example, both $121$ and $211$ have this property.
1. A) $226$
2. B) $243$
3. C) $270$
4. D) $469$
5. E) $486$
56.2013 AMC 10A #25
All 20 diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect?
1. A) $49$
2. B) $65$
3. C) $70$
4. D) $96$
5. E) $128$
57.2012 AMC 10A #23
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?
1. A) $60$
2. B) $170$
3. C) $290$
4. D) $320$
5. E) $660$

Basic Counting — answersAMC 10 Step by Step

Name: Date: October 28, 2026

Keep this page separate from the problems.

Answers

#	Answer	The idea it turns on
1	B	The person in the middle is counted once from each end. Count the people on each side of that person, then add that person back once.
2	B	The value only depends on which two digits get multiplied together. There are just three ways to split 1, 2, 3, 4 into two pairs.
3	A	Count how many 2-inch pieces fit along each side of the pan, 10 along one side and 9 along the other, then multiply.
4	D	Count all subsets, then take away the ones with no prime at all, which are the subsets made only from 4, 6, 8, 9.
5	E	Each time a 2-point basket becomes a 3-point basket, the total goes up by exactly 1. Six choices for the number of 3-pointers give six totals.
6	C	Rows 12 through 22 means both end rows count, so there are 11 rows, not 10. Then multiply 11 rows by 33 seats.
7	B	It is easier to count the two-digit numbers with no 7 at all (8 choices for the tens digit, 9 for the ones digit) and take them away from 90.
8	B	Split the juniors into three pairs (15 ways), split the seniors into three pairs (15 ways), then match each junior pair with a senior pair (6 ways).
9	D	Follow one vertex's number: 3 edges use it and each edge sits on 2 faces, so it is counted 6 times. The cube's value is 6 times 21.
10	D	Whatever the first two digits are, exactly 5 of the 10 possible last digits make the count of even digits odd. So exactly half of all 900 three-digit numbers work.
11	B	The last digit must be 0 (even and divisible by 5). The first digit has 4 choices, since it cannot be 0, and each middle digit has 5.
12	B	Place the colors one group at a time: choose 3 of the 7 spots for yellow, then 2 of the remaining 4 for green, then order brown and purple.
13	B	A pair shakes hands exactly when at least one of the two is one of the 10 strangers. Count stranger-with-group pairs, then stranger-with-stranger pairs.
14	D	Count the flat toothpicks row by row, then use the fact that there are just as many standing ones. The 3-step staircase has 18, the 5-step one has 40.
15	A	Ten ways to pick 2 people means the council has 5 members. Then count the ways to pick 3 of those 5, which is also 10.
16	C	English is always in, so pick 3 more from the other 5 courses. Of those 10 choices, only History, Art, Latin has no math, so 9 are allowed.
17	A	The first digit must be 2 or 5 and the last digit must be 2 or 8. If the first digit is 2, the last digit can only be 8.
18	A	The middle digit has no partner, so it must be the 5 that appears an odd number of times. The first three digits are 2, 3, 5 in any order.
19	C	Only the single plus sign can be moved earlier. It can join the 3 with the left product, the right product, both, or neither, giving 26, 46, 50 and 70.
20	D	Each tourist has 2 choices, giving 64 ways with no rule. Only 2 of them leave a guide with nobody, so the answer is 62.
21	D	Once a and c are picked, b is their average, so it is fixed. It is a whole number exactly when a and c are both even or both odd.
22	C	For each of the 8 condiments you just say yes or no, giving 2 to the 8th = 256 choices. Multiply by 3 patty choices.
23	D	Count dance pairs two ways. The men make 12 times 3 = 36 pairs; each woman is in 2 pairs, so there are 18 women.
24	C	Old plates: 26 times 10 four times. New plates: 26 three times times 10 three times. Divide new by old and cancel: 26 squared over 10.
25	E	There are 6 times m times m dinners, which must reach 365. Seven main courses give 294, too few; eight give 384.
26	D	Count the words that come before USAMO in blocks: first all words starting with A, M, O or S, then the words starting UA, UM or UO. USAMO is next.
27	C	Add the inequalities in pairs to trap x, y and z between 0 and 2. Then list the few triples, sorted by their sum.
28	D	Sort digits 1 to 9 into groups: 2 is even and prime, 4, 6, 8 only even, 3, 5, 7 only prime, 1 and 9 neither. Then count two cases.
29	A	All students speak the same number of languages, so the rule just says no two students speak exactly the same group. Choosing 4 of 9 languages gives 126 groups.
30	E	Picking the digits decides the number, so count sets of digits. The digit 0 can end a downno but can never start an upno.
31	E	Both probabilities share the bottom 5 to the 20th, so compare counts. Most cancels, leaving 20 ways to pick the special bins times 4/5.
32	C	Curves meet only if the steeper parabola has the smaller constant. Swapping the two constants turns each good choice into a bad one, so half of all choices work.
33	D	Stack all 9 cubes and then take off the top one. Every 8-cube tower comes from exactly one 9-cube stack, so just count 9-cube stacks.
34	B	Choosing a set of different digits gives exactly one increasing number and one decreasing number. Count digit sets: 511 increasing, 1022 decreasing, minus the 9 one-digit numbers counted twice.
35	C	Count in nickels, with a dime worth 2. With at least one nickel, every amount up to the total can be made, so the total is 17 nickels.
36	A	The time is right only when no 1 appears. Multiply the share of good hours (8 of 12) by the share of good minutes (45 of 60).
37	C	Choose which 3 rows (10 ways) and which 3 columns (10 ways) the blocks use, then decide which row goes with which column (6 ways): 10 times 10 times 6.
38	C	Glue the two letters into one block. It can start in 5 of the 6 spots; fill it with 26 × 26 letter pairs and the rest with 10^4 digits.
39	E	Count the opposite: four-digit numbers with no 2 and no 3. There are 7 × 8 × 8 × 8 = 3584 of them; subtract from 9000.
40	D	Seven different digits in increasing order can be arranged only one way. So a phone number is fixed by which of the eight digits 2 through 9 is left out.
41	C	Each of the five digits sits in each place equally often, so every place averages (1+3+5+7+8)/5 = 4.8. The average number is 4.8 times 11111.
42	D	Count by size. The numbers of squares that cover the center are 1, 4, 9, 4, 1 for sizes 1 through 5, which adds to 19.
43	A	Line up the 16 people, cut the line into four groups of 4, and divide out orders that do not matter. Then count the 3s on top and bottom.
44	D	Two passes means cards 1 to k appear in increasing order and so do cards k+1 to 13. Choosing which k spots hold the first group decides the whole row.
45	E	Each side of the polygon with fewer sides crosses the other polygon exactly twice, so each pair meets at twice the smaller side count.
46	E	Every route goes: top face, along the top ring, down, along the bottom ring, bottom face. Each ring walk has 9 options, so 5 times 9 times 2 times 9.
47	C	Split into cases by how many pairs are 'across' pairs. Everyone left must pair with a neighbor, so each leftover stretch of the circle needs an even number of people.
48	C	Each move sends corner A to a neighboring corner, so after 19 moves the square is exactly one move from home, and exactly one 20th move finishes.
49	B	Count all ways to pick 3 of the 25 points, then subtract picks lying on one line. Sort those lines by direction so none are missed.
50	B	Give each song a label: the group of girls who like it. There are 7 allowed labels, and the three two-girl labels must each be used at least once.
51	A	Each inside triangle comes from three chords that all cross, using six different points; each group of 6 points gives exactly one triangle.
52	B	Each letter is in both subsets, only the first, or only the second. Choose the 2 shared letters, split the other 3, then halve.
53	C	Every toothpick is a side of exactly one upward-pointing small triangle. Count those, 1 + 2 + ... + 1002, and multiply by 3.
54	C	A fair number's digits climb to the largest digit, then fall. Once the digits are chosen, every digit except the largest just picks the left or right side.
55	A	A three-digit number is a multiple of 11 when first minus middle plus last is 0 or 11. So one digit is the others' sum, or that sum minus 11.
56	A	Any 4 corners give one crossing, so 70 in all, but the center and eight other points have extra diagonals through them and are overcounted.
57	B	Everyone has 1 to 4 friends. Swapping friends and non-friends turns 1 into 4 and 2 into 3, so count only 1 and 2 friends, then double.

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.