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 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 squares. A

scanning code is called if its look does not change when

the entire square is rotated by a multiple of 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?

[Q.2]In the expression each blank is to be

filled in with one of the digits or with each digit being

used once. How many different values can be obtained?

[Q.3]How many subsets of contain at least one prime

number?

[Q.4]A list ofpositive integers has a unique mode, which occurs

exactlytimes. What is the least number of distinct values that

can occur in the list?

[Q.5]At a gathering ofpeople, there are people who all know each

other andpeople 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?

[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?

[Q.7]How many integers betweenand , inclusive, have the

property that some permutation of its digits is a multiple of

betweenand ?

For example, bothand have this property.

[Q.8]There arestudents 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 arestudents taking yoga, taking bridge, and

taking painting. There are $9$ students taking at least two classes.

How many students are taking all three classes?

[Q.9]How many of the base-ten numerals for the positive integers less

than or equal tocontain the digit ?

[Q.10]In the figure below,of the disks are to be painted blue,

are to be painted red, andis 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?

[Q.11]How many of the base-ten numerals for the positive integers less

than or equal tocontain the digit ?

[Q.12]Each vertex of a cube is to be labeled with an integerthrough

,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?

[Q.13]Five friends sat in a movie theater in a row containingseats,

numberedto $5$ 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?

[Q.14]Erin the ant starts at a given corner of a cube and crawls along

exactlyedges 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?

[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?

[Q.16]The numbersare to be arranged in a circle.

An arrangement isif it is not true that for every from

to one can find a subset of the numbers that appear

consecutively on the circle that sum to. Arrangements that

differ only by a rotation or a reflection are considered the same.

How many different bad arrangements are there?

[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?

[Q.18]A student council must select a two-person welcoming committee

and a three-person planning committee from among its members. There

are exactlyways 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?

[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?

[Q.20]Alldiagonals 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?

[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?

[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?

[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?

[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?

[Q.25]Let (, $a_2$, ... $a_{10}$) be a list of the first 10 positive

integers such that for eacheither

or or

both appear somewhere before

in the list. How many such lists are there?

[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?

[Q.27]How many even integers are there between 200 and 700 whose

digits are all different and come from the set?

[Q.28]Each vertex of convex pentagonis to be assigned a

color. There arecolors to choose from, and the ends of each

diagonal must have different colors. How many different colorings are

possible?

[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?

[Q.30]Two subsets of the setare to be chosen so that

their union isand 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?

[Q.31]Ten chairs are evenly spaced around a round table and numbered

clockwise fromthrough . 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?

Also Visit: Math Olympiad Program

## AMC 12

[Q.32]How many subsets ofcontain at least one prime

number?

[Q.33]A set ofpeople participate in an online video basketball

tournament. Each person may be a member of any number of-player

teams, but no two teams may have exactly the samemembers. The site

statistics show a curious fact: The average, over all subsets

of size<of the set of participants, of the number of complete

teams whose members are among thosepeople is equal to the

reciprocal of the average, over all subsets of sizeof the

set ofparticipants, of the number of complete teams whose members

are among thosepeople. How many values , , can

be the number of participants?

[Q.34]A set of teams held a round-robin tournament in which every team

played every other team exactly once. Every team wongames and

lostgames; there were no ties. How many sets of three teams

were there in which beat , beat , and beat

[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?$

[Q.36]A fancy bed and breakfast inn hasrooms, each with a

distinctive color-coded decor. One dayfriends 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 thanfriends per room.

In how many ways can the innkeeper assign the guests to the rooms?$

[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?$

[Q.38]Letbe a list of the first 10 positive integers

such that for eacheither or or both

appear somewhere beforein the list. How many such lists are there$ ?

[Q.39]How many positive integers less thanare times the sum of

their digits$ ?

[Q.40]Ten women sit inseats in a line. All of the get up and

then reseat themselves using allseats, 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$ ?