Combinatorics is one of the most important topic for the preparation of Mathematics Olympiad culture as well as American Mathematical Contest(also known as AMC). AMC10/12 Combinatorics Problem which is composed of selected problems from previous year.

AMC 10

[Q.1]A scanning code consists of a $latex 7 \times 7$ 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 $latex 49$ squares. A
 scanning code is called $latex \text{symmetric}$ if its look does not change when
 the entire square is rotated by a multiple of $latex 90 ^{\circ}$ 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?

$latex \textbf{(A)} \text{ 510} \qquad \textbf{(B)} \text{ 1022} \qquad \textbf{(C)} \text{ 8190} \qquad \textbf{(D)} \text{ 8192} \qquad \textbf{(E)} \text{ 65,534}$

[Q.2]In the expression $latex \left(\underline{\qquad}\times\underline{\qquad}\right)+\left(\underline{\qquad}\times\underline{\qquad}\right)$ each blank is to be
 filled in with one of the digits $latex 1,2,3,$ or $latex 4,$ with each digit being
 used once. How many different values can be obtained?

$latex \textbf{(A) }2 \qquad \textbf{(B) }3\qquad \textbf{(C) }4 \qquad \textbf{(D) }6 \qquad \textbf{(E) }24 \qquad$

[Q.3]How many subsets of $latex \{2,3,4,5,6,7,8,9\}$ contain at least one prime
 number?

$latex \textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$

[Q.4]A list of $latex 2018$ positive integers has a unique mode, which occurs
 exactly $latex 10$ times. What is the least number of distinct values that
 can occur in the list?

$latex \textbf{(A)}\ 202\qquad\textbf{(B)}\ 223\qquad\textbf{(C)}\ 224\qquad\textbf{(D)}\ 225\qquad\textbf{(E)}\ 234$

[Q.5]At a gathering of $latex 30$ people, there are $latex 20$ people who all know each
 other and $latex 10$ 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 within the group?

$latex \textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490$

[Q.6]Alice refuses to sit next to either Bob or Carla. Derek refuses
 to sit next to Eric. How many ways are there for the five of them to
 sit in a row of 5 chairs under these conditions?

$latex \textbf{(A)}\ 12\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 40$

[Q.7]How many integers between $latex 100$ and $latex 999$, inclusive, have the
 property that some permutation of its digits is a multiple of $latex 11$
 between $latex 100$ and $latex 999$?
 For example, both $latex 121$ and $latex 211$ have this property.

$latex \mathrm{\textbf{(A)} \ }226\qquad \mathrm{\textbf{(B)} \ } 243 \qquad \mathrm{\textbf{(C)} \ } 270 \qquad \mathrm{\textbf{(D)} \ }469\qquad \mathrm{\textbf{(E)} \ } 486$

[Q.8]There are $latex 20$ students participating in an after-school program
 offering classes in yoga, bridge, and painting. Each student must
 take at least one of these three classes, but may take two or all
 three. There are $latex 10$ students taking yoga, $latex 13$ taking bridge, and $latex 9$
 taking painting. There are $9$ students taking at least two classes.
 How many students are taking all three classes?

$latex \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $

[Q.9]How many of the base-ten numerals for the positive integers less
 than or equal to $latex 2017$ contain the digit $latex 0$?

$latex \textbf{(A)}\ 469\qquad\textbf{(B)}\ 471\qquad\textbf{(C)}\ 475\qquad\textbf{(D)}\ 478\qquad\textbf{(E)}\ 481 $

[Q.10]In the figure below, $latex 3$ of the $latex 6$ disks are to be painted blue, $latex 2$ 
 are to be painted red, and $latex 1$ is to be painted green. Two paintings
that can be obtained from one another by a rotation or a reflection
 of the entire figure are considered the same. How many different
 paintings are possible?

$latex \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 15 $

[Q.11]How many of the base-ten numerals for the positive integers less
 than or equal to $latex 2017$ contain the digit $latex 0$?

$latex \textbf{(A)}\ 1024\qquad\textbf{(B)}\ 1524\qquad\textbf{(C)}\ 1533\qquad\textbf{(D)}\ 1536\qquad\textbf{(E)}\ 2048 $

[Q.12]Each vertex of a cube is to be labeled with an integer $latex 1$ through
 $latex 8$,with each integer being used once, in such a way that the sum of
 the four numbers on the vertices of a face is the same for each face.
 Arrangements that can be obtained from each other through rotations
 of the cube are considered to be the same. How many different
 arrangements are possible?

$latex \textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24 $

[Q.13]Five friends sat in a movie theater in a row containing $latex 5$ seats,
 numbered $latex 1$ to $5$ from left to right. (The directions "left" and
 "right" are from the point of view of the people as they sit in the
 seats.) During the movie Ada went to the lobby to get some popcorn.
 When she returned, she found that Bea had moved two seats to the
 right, Ceci had moved one seat to the left, and Dee and Edie had
 switched seats, leaving an end seat for Ada. In which seat had Ada
 been sitting before she got up?

$latex \textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4\qquad \textbf{(E) } 5 $

[Q.14]Erin the ant starts at a given corner of a cube and crawls along
 exactly $latex 7$ edges in such a way that she visits every corner exactly
 once. and then finds that she is unable to return along an edge to her
 starting point. How many paths are there meeting these conditions?

$latex \textbf{(A) }\text{6}\qquad\textbf{(B) }\text{9}\qquad\textbf{(C) }\text{12}\qquad\textbf{(D) }\text{18}\qquad\textbf{(E) }\text{24} $

[Q.15]Walking down Jane Street, Ralph passed four houses in a row,
 each painted a different color. He passed the orange house before the
 red house, and he passed the blue house before the yellow house. The
 blue house was not next to the yellow house. How many orderings
 of the colored houses are possible?

$latex \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 $

[Q.16]The numbers $latex 1, 2, 3, 4, 5$ are to be arranged in a circle.
 An arrangement is $latex \textit{bad}$ if it is not true that for every $latex n$ from
 $latex 1$ to $latex 15$ one can find a subset of the numbers that appear
 consecutively on the circle that sum to $latex n$. Arrangements that
 differ only by a rotation or a reflection are considered the same.
 How many different bad arrangements are there?

$latex \textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5 $

[Q.17]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?

$latex \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 16 $

[Q.18]A student council must select a two-person welcoming committee
 and a three-person planning committee from among its members. There
 are exactly $latex 10$ 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?

$latex \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 25 $

[Q.19]Central High School is competing against Northern High School
 in a backgammon match. Each school has three players, and the contest
 rules require that each player play two games against each of the
 other school's players. The match takes place in six rounds, with
 three games played simultaneously in each round. In how many different
 ways can the match be scheduled?

$latex \textbf{(A)}\ 540\qquad\textbf{(B)}\ 600\qquad\textbf{(C)}\ 720\qquad\textbf{(D)}\ 810\qquad\textbf{(E)}\ 900 $

[Q.20]All $latex 20$ 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?

$latex \textbf{(A)}\ 49\qquad\textbf{(B)}\ 65\qquad\textbf{(C)}\ 70\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 128 $

[Q.21]Chubby makes nonstandard checkerboards that have $31$ squares on
 each side. The checkerboards have a black square in every corner and
 alternate red and black squares along every row and column. How many
 black squares are there on such a checkerboard?

$latex \textbf{(A)}\ 480 \qquad\textbf{(B)}\ 481 \qquad\textbf{(C)}\ 482 \qquad\textbf{(D)}\ 483 \qquad\textbf{(E)}\ 484 $

[Q.22]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?

$latex \text{(A)}\ 60\qquad\text{(B)}\ 170\qquad\text{(C)}\ 290\qquad\text{(D)}\ 320\qquad\text{(E)}\ 660 $

[Q.23]A dessert chef prepares the dessert for every day of a week
 starting with Sunday. The dessert each day is either cake, pie, ice
 cream, or pudding. The same dessert may not be served two days in a
 row. There must be cake on Friday because of a birthday. How many
 different dessert menus for the week are possible?

$latex \textbf{(A)}\ 729\qquad\textbf{(B)}\ 972\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 2187\qquad\textbf{(E)}\ 2304 $

[Q.24]In a round-robin tournament with 6 teams, each team plays one
 game against each other team, and each game results in one team
 winning and one team losing. At the end of the tournament, the teams
 are ranked by the number of games won. What is the maximum number of
 teams that could be tied for the most wins at the end on the
 tournament?

$latex \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 $

[Q.25]Let ($latex a_1$, $a_2$, ... $a_{10}$) be a list of the first 10 positive
 integers such that for each $latex 2\le$ $latex i$ $latex \le10$ either
 $latex a_i + 1$ or $latex a_i-1$ or
 both appear somewhere before
 $latex a_i$ in the list. How many such lists are there?

$latex \textbf{(A)}\ \ 120\qquad\textbf{(B)}\ 512\qquad\textbf{(C)}\ \ 1024\qquad\textbf{(D)}\ 181,440\qquad\textbf{(E)}\ \ 362,880 $

[Q.26]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 two girls but disliked by the third. In how many
 different ways is this possible?

$latex \textbf{(A)}\ 108\qquad\textbf{(B)}\ 132\qquad\textbf{(C)}\ 671\qquad\textbf{(D)}\ 846\qquad\textbf{(E)}\ 1105 $

[Q.27]How many even integers are there between 200 and 700 whose
 digits are all different and come from the set $latex \{1,2,5,7,8,9\}$ ?

$latex \text{(A)}\,12 \qquad\text{(B)}\,20 \qquad\text{(C)}\,72 \qquad\text{(D)}\,120 \qquad\text{(E)}\,200 $

[Q.28]Each vertex of convex pentagon $latex ABCDE$ is to be assigned a
 color. There are $latex 6$ colors to choose from, and the ends of each
 diagonal must have different colors. How many different colorings are
 possible?

$latex \textbf{(A)}\ 2520\qquad\textbf{(B)}\ 2880\qquad\textbf{(C)}\ 3120\qquad\textbf{(D)}\ 3250\qquad\textbf{(E)}\ 3750 $

[Q.29]Seven distinct pieces of candy are to be distributed among three
 bags. The red bag and the blue bag must each receive at least one piece
 of candy; the white bag may remain empty. How many arrangements are
 possible?

$latex \textbf{(A)}\ 1930 \qquad \textbf{(B)}\ 1931 \qquad \textbf{(C)}\ 1932 \qquad \textbf{(D)}\ 1933 \qquad \textbf{(E)}\ 1934 $

[Q.30]Two subsets of the set $latex S=\lbrace a,b,c,d,e\rbrace$ are to be chosen so that
 their union is $latex S$ 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?

$latex \mathrm{(A)}\ 20\qquad\mathrm{(B)}\ 40\qquad\mathrm{(C)}\ 60\qquad\mathrm{(D)}\ 160\qquad\mathrm{(E)}\ 320 $

[Q.31]Ten chairs are evenly spaced around a round table and numbered
 clockwise from $latex 1$ through $latex 10$. Five married couples are to sit in the
 chairs with men and women alternating, and no one is to sit either
 next to or across from his/her spouse. How many seating
 arrangements are possible?

$latex \mathrm{(A)}\ 240\qquad\mathrm{(B)}\ 360\qquad\mathrm{(C)}\ 480\qquad\mathrm{(D)}\ 540\qquad\mathrm{(E)}\ 720 $

Also Visit: Math Olympiad Program

AMC 12

[Q.32]How many subsets of  $latex \{2,3,4,5,6,7,8,9\}$ contain at least one prime
 number?

$latex (\text{A}) \indent 128 \qquad (\text{B}) \indent 192  \qquad (\text{C}) \indent 224  \qquad (\text{D}) \indent 240 \qquad (\text{E}) \indent 256 $

[Q.33]A set of $latex n$ people participate in an online video basketball
 tournament. Each person may be a member of any number of $latex 5$-player
 teams, but no two teams may have exactly the same $latex 5$ members. The site
 statistics show a curious fact: The average, over all subsets
 of size< $latex 9$ of the set of $latex n$ participants, of the number of complete
 teams whose members are among those $latex 9$ people is equal to the
 reciprocal of the average, over all subsets of size $latex 8$ of the
 set of $latex n$ participants, of the number of complete teams whose members
 are among those $latex 8$ people. How many values $latex n$,  $latex 9\leq n\leq 2017$, can
 be the number of participants?

$latex \textbf{(A) } 477 \qquad \textbf{(B) } 482 \qquad \textbf{(C) } 487 \qquad \textbf{(D) } 557 \qquad \textbf{(E) } 562$

[Q.34]A set of teams held a round-robin tournament in which every team
 played every other team exactly once. Every team won $latex 10$ games and
 lost $latex 10$ games; there were no ties. How many sets of three teams
 $latex \{A, B, C\}$ were there in which $latex A$ beat $latex B$, $latex B$ beat $latex C$, and $latex C$ beat $latex A?$

$latex \textbf{(A)}\ 385 \qquad \textbf{(B)}\ 665 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 1140 \qquad \textbf{(E)}\ 1330$

[Q.35]Six chairs are evenly spaced around a circular table. One person
 is seated in each chair. Each person gets up and sits down in a chair
 that is not the same chair and is not adjacent to the chair he or she
 originally occupied, so that again one person is seated in each chair.
 In how many ways can this be done?$

$latex \textbf{(A)}\; 14 \qquad\textbf{(B)}\; 16 \qquad\textbf{(C)}\; 18 \qquad\textbf{(D)}\; 20 \qquad\textbf{(E)}\; 24$

[Q.36]A fancy bed and breakfast inn has $latex 5$ rooms, each with a
 distinctive color-coded decor. One day $latex 5$ friends arrive to spend
 the night. There are no other guests that night. The friends can room in
 any combination they wish, but with no more than $latex 2$ friends per room.
 In how many ways can the innkeeper assign the guests to the rooms?$

$latex \textbf{(A) }2100\qquad \textbf{(B) }2220\qquad \textbf{(C) }3000\qquad \textbf{(D) }3120\qquad \textbf{(E) }3125\qquad$

[Q.37]Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie,
 and Cotton-tail. These five rabbits are to be distributed to four
 different pet stores so that no store gets both a parent and a child.
 It is not required that every store gets a rabbit. In how many
 different ways can this be done?$

$latex \textbf{(A)} \ 96 \qquad  \textbf{(B)} \ 108 \qquad  \textbf{(C)} \ 156 \qquad  \textbf{(D)} \ 204 \qquad  \textbf{(E)} \ 372$

[Q.38]Let $latex (a_1,a_2, \dots ,a_{10})$ be a list of the first 10 positive integers
 such that for each $latex 2 \le i \le 10$ either $latex a_i+1$ or $latex a_i-1$ or both
 appear somewhere before $latex a_i$ in the list. How many such lists are there$ ?

$latex \textbf{(A)}\ 120\qquad\textbf{(B)}\ 512\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 181,440\qquad\textbf{(E)}\ 362,880$

[Q.39]How many positive integers less than $latex 1000$ are $latex 6$ times the sum of
 their digits$ ?

$latex \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 12$

[Q.40]Ten women sit in $latex 10$ seats in a line. All of the $latex 10$ get up and
 then reseat themselves using all $latex 10$ seats, each sitting in the
 seat she was in before or a seat next to the one she occupied before.
 In how many ways can the women be reseated$ ?

$latex \textbf{(A)}\ 89\qquad \textbf{(B)}\ 90\qquad \textbf{(C)}\ 120\qquad \textbf{(D)}\ 2^{10}\qquad \textbf{(E)}\ 2^2 3^8$