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Technique: complementary counting

Count the complement and subtract from the total

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Technique: complementary countingAMC 10 Step by Step

Name: Date: September 13, 2026

12 problems · 40 minutes · 6 / 1.5 / 0

1.2003 AMC 10A #15
What is the probability that an integer in the set $\{1,2,3,...,100\}$ is divisible by $2$ and not divisible by $3$ ?
1. A)
  $\frac{1}{6}$
2. B)
  $\frac{33}{100}$
3. C)
  $\frac{17}{50}$
4. D)
  $\frac{1}{2}$
5. E)
  $\frac{18}{25}$
2.2013 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)
  $16$
3.2016 AMC 10B #12
Two different numbers are selected at random from $( 1, 2, 3, 4, 5)$ and multiplied together. What is the probability that the product is even?
1. A)
  $0.2$
2. B)
  $0.4$
3. C)
  $0.5$
4. D)
  $0.7$
5. E)
  $0.8$
4.2017 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)
  $490$
5.2017 AMC 10B #16
How many of the base-ten numerals for the positive integers less than or equal to $2017$ contain the digit $0$ ?
1. A)
  $469$
2. B)
  $471$
3. C)
  $475$
4. D)
  $478$
5. E)
  $481$
6.2021 AMC Fall 10B #14
Una rolls $6$ standard $6$ -sided dice simultaneously and calculates the product of the $6{ }$ numbers obtained. What is the probability that the product is divisible by $4?$
1. A)
  $\frac34$
2. B)
  $\frac{57}{64}$
3. C)
  $\frac{59}{64}$
4. D)
  $\frac{187}{192}$
5. E)
  $\frac{63}{64}$
7.2023 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)
  $511$
8.2005 AMC 10B #22
For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 + 2 + \cdots + n?$
1. A)
  $8$
2. B)
  $12$
3. C)
  $16$
4. D)
  $17$
5. E)
  $21$
9.2016 AMC 10A #17
Let $N$ be a positive multiple of $5$ . One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$ ?
1. A)
  $12$
2. B)
  $14$
3. C)
  $16$
4. D)
  $18$
5. E)
  $20$
10.2016 AMC 10B #22
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$ , $B$ beat $C$ , and $C$ beat $A?$
1. A)
  $385$
2. B)
  $665$
3. C)
  $945$
4. D)
  $1140$
5. E)
  $1330$
11.2021 AMC Fall 10B #23
Each of the $5{ }$ sides and the $5{ }$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color?
1. A)
  $\frac23$
2. B)
  $\frac{105}{128}$
3. C)
  $\frac{125}{128}$
4. D)
  $\frac{253}{256}$
5. E)
  $1$
12.2021 AMC 10A #23
Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up", the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops?
1. A)
  $\frac{9}{16}$
2. B)
  $\frac{5}{8}$
3. C)
  $\frac{3}{4}$
4. D)
  $\frac{25}{32}$
5. E)
  $\frac{13}{16}$

Technique: complementary counting — answersAMC 10 Step by Step

Name: Date: September 13, 2026

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Answers

#	Answer	Topic and the idea it turns on
1	C	Divisibility & Factors — Count the 50 even numbers, then remove the 16 multiples of 6; 34 of 100 remain.
2	C	Basic Counting — English is forced, so choose 3 of the other 5 courses and throw away the single choice with no math course.
3	D	Basic Probability — A product is odd only when both factors are odd; 3 of the 10 pairs are odd-odd, so 7 of 10 products are even.
4	B	Basic Counting — A handshake happens exactly when at least one of the pair is a stranger: 10 times 20 mixed pairs plus C(10,2) stranger-stranger pairs.
5	A	Bases & Digits — Count numbers with no zero digit (9, 81, 729 for 1 to 3 digits, 729 for 1000-1999, none for 2000-2017) and subtract from 2017.
6	C	Basic Probability — The product misses a factor of 4 only if all six dice are odd, or exactly one die shows 2 or 6 and the rest are odd.
7	E	Basic Counting — An upno is a subset of {1,...,9} of size at least 2 and a downno a subset of {0,...,9}; the counts are 2^9 - 10 and 2^10 - 11.
8	C	Divisibility & Factors — n!/(1+...+n) = 2(n-1)!/(n+1), which fails to be an integer exactly when n+1 is an odd prime; eight such n up to 24, so 24 - 8 = 16.
9	A	Basic Probability — Only the red ball's position matters; with N = 5m it fails exactly when 2m+1 to 3m-1 greens lie to its left, so P(N) = 1 - (m-1)/(5m+1).
10	A	Games & Processes — There are 21 teams; a triple is non-cyclic iff one team beat the other two, and each team supplies C(10,2) such triples, so C(21,3) - 21*45 = 385.
11	D	Basic Probability — With no one-color triangle, every vertex has exactly 2 red and 2 blue edges, so the red edges form a 5-cycle; only 12 of 1024 colorings.
12	D	Conditional Probability & States — Only three states matter: center, edge, corner. From an edge, half the hops reach a corner, a quarter return to the center, a quarter wrap to another edge.