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