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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

20 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)

1.2007 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) $25$
2.2004 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) $120,960$
3.2004 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) $24$
4.2003 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) $\frac{26^3}{10^2}$
5.2003 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) $8$
6.2002 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) $116$
7.2025 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) $17$
8.2025 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) $608$
9.2024 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) $100$
10.2023 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) $511$
11.2021 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) $16$
12.2021 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) $360$
13.2019 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) $40,320$
14.2017 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) $2048$
15.2015 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) $7$
16.2009 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) $\frac 9{10}$
17.2007 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) $3600$
18.2006 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) $5\times 10^3\times 26^3$
19.2006 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) $5416$
20.2005 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) $9$

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	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.
2	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.
3	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.
4	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.
5	E	There are 6 times m times m dinners, which must reach 365. Seven main courses give 294, too few; eight give 384.
6	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.
7	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.
8	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.
9	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.
10	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.
11	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.
12	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.
13	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.
14	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.
15	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.
16	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).
17	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.
18	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.
19	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.
20	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.

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.