Difference between revisions of "2013 AMC 10A Problems/Problem 24"
(→Solution 1) |
|||
(3 intermediate revisions by 2 users not shown) | |||
Line 12: | Line 12: | ||
Round 1: <math>AX</math> <math>BY</math> <math>CZ</math> | Round 1: <math>AX</math> <math>BY</math> <math>CZ</math> | ||
+ | |||
Round 2: <math>AX</math> <math>BZ</math> <math>CY</math> | Round 2: <math>AX</math> <math>BZ</math> <math>CY</math> | ||
+ | |||
Round 3: <math>AY</math> <math>BX</math> <math>CZ</math> | Round 3: <math>AY</math> <math>BX</math> <math>CZ</math> | ||
+ | |||
Round 4: <math>AY</math> <math>BZ</math> <math>CX</math> | Round 4: <math>AY</math> <math>BZ</math> <math>CX</math> | ||
+ | |||
Round 5: <math>AZ</math> <math>BX</math> <math>CY</math> | Round 5: <math>AZ</math> <math>BX</math> <math>CY</math> | ||
+ | |||
Round 6: <math>AZ</math> <math>BY</math> <math>CX</math> | Round 6: <math>AZ</math> <math>BY</math> <math>CX</math> | ||
+ | |||
The above mentioned schedule ensures that each player of one team plays twice with each player from another team. Now you can generate a completely new schedule by permutating those <math>6</math> rounds and that can be done in <math>6!=720</math> ways. | The above mentioned schedule ensures that each player of one team plays twice with each player from another team. Now you can generate a completely new schedule by permutating those <math>6</math> rounds and that can be done in <math>6!=720</math> ways. | ||
Line 23: | Line 29: | ||
(a) | (a) | ||
+ | |||
Round 1: <math>AX</math> <math>BZ</math> <math>CY</math> | Round 1: <math>AX</math> <math>BZ</math> <math>CY</math> | ||
+ | |||
Round 2: <math>AX</math> <math>BZ</math> <math>CY</math> | Round 2: <math>AX</math> <math>BZ</math> <math>CY</math> | ||
+ | |||
Round 3: <math>AY</math> <math>BX</math> <math>CZ</math> | Round 3: <math>AY</math> <math>BX</math> <math>CZ</math> | ||
+ | |||
Round 4: <math>AY</math> <math>BX</math> <math>CZ</math> | Round 4: <math>AY</math> <math>BX</math> <math>CZ</math> | ||
+ | |||
Round 5: <math>AZ</math> <math>BY</math> <math>CX</math> | Round 5: <math>AZ</math> <math>BY</math> <math>CX</math> | ||
+ | |||
Round 6: <math>AZ</math> <math>BY</math> <math>CX</math> | Round 6: <math>AZ</math> <math>BY</math> <math>CX</math> | ||
(b) | (b) | ||
+ | |||
Round 1: <math>AX</math> <math>BY</math> <math>CZ</math> | Round 1: <math>AX</math> <math>BY</math> <math>CZ</math> | ||
+ | |||
Round 2: <math>AX</math> <math>BY</math> <math>CZ</math> | Round 2: <math>AX</math> <math>BY</math> <math>CZ</math> | ||
+ | |||
Round 3: <math>AY</math> <math>BZ</math> <math>CX</math> | Round 3: <math>AY</math> <math>BZ</math> <math>CX</math> | ||
+ | |||
Round 4: <math>AY</math> <math>BZ</math> <math>CX</math> | Round 4: <math>AY</math> <math>BZ</math> <math>CX</math> | ||
+ | |||
Round 5: <math>AZ</math> <math>BX</math> <math>CY</math> | Round 5: <math>AZ</math> <math>BX</math> <math>CY</math> | ||
+ | |||
Round 6: <math>AZ</math> <math>BX</math> <math>CY</math> | Round 6: <math>AZ</math> <math>BX</math> <math>CY</math> | ||
Line 61: | Line 79: | ||
There are <math>1+1+2\cdot2\cdot2\cdot1 = 10</math> ways to assign opponents to <math>B</math> and <math>90\cdot10 = 900</math> ways to order the games. <math>\boxed{\textbf{(E)} 900}</math> | There are <math>1+1+2\cdot2\cdot2\cdot1 = 10</math> ways to assign opponents to <math>B</math> and <math>90\cdot10 = 900</math> ways to order the games. <math>\boxed{\textbf{(E)} 900}</math> | ||
+ | |||
+ | == Solution 3(similar to 2) == | ||
+ | We only need to worry about what the other member of the pair would be for our own competitors. If our competitors are <math>A,B,C</math> and the others are <math>1,2,3</math>, then we only need to arrange the numbers <math>1,1,2,2,3,3</math> such that we don't have the same number in the same spots. For our first row of numbers, or our first player, they can be arranged in any of <cmath>\frac{6!}{2!2!2!}</cmath> ways as nothing has happened yet. This contributes <math>90</math> to our final product. Once we determine our second row, or player, the third will immediately follow thus we only need to figure out the second one. WLOG, our first arrangement was <math>112233</math>. For our <math>1</math>'s, we can put them anywhere that is not in our first <math>2</math> spots. If we put both of them in the same "region", like the <math>2</math>'s or the <math>3</math>'s, then it will give a different case than if they were in different regions. | ||
+ | |||
+ | Case 1 : They are in the same region | ||
+ | |||
+ | There are <math>2</math> ways that the <math>1</math>'s can be in the same region. If, let's say its in the <math>2</math>'s region, we try placing the <math>2's</math>, we can only place them in the <math>3</math> region in order to fit the <math>3</math>'s. Therefore, this entire case produces <math>2</math> arrangements. \\ | ||
+ | |||
+ | Case 2: They are in a different region | ||
+ | |||
+ | There are <math>4</math> ways that the <math>1</math>'s can be in different regions. For the <math>2</math>'s and <math>3</math>'s, they will have to go in each other's region with <math>1</math> and the other can be arranged in <math>2</math> ways in the <math>1</math> region giving a total of <math>8</math> for this case.\\ | ||
+ | |||
+ | Finally, <math>90(2+8) = 900</math> | ||
+ | |||
+ | ~Vedoral | ||
==Graph Theory Insight== | ==Graph Theory Insight== |
Latest revision as of 20:13, 15 July 2024
Contents
Problem
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?
Solution 1
Let us label the players of the first team ,
, and
, and those of the second team,
,
, and
.
. One way of scheduling all six distinct rounds could be:
Round 1:
Round 2:
Round 3:
Round 4:
Round 5:
Round 6:
The above mentioned schedule ensures that each player of one team plays twice with each player from another team. Now you can generate a completely new schedule by permutating those rounds and that can be done in
ways.
. One can also make the schedule in such a way that two rounds are repeated.
(a)
Round 1:
Round 2:
Round 3:
Round 4:
Round 5:
Round 6:
(b)
Round 1:
Round 2:
Round 3:
Round 4:
Round 5:
Round 6:
As mentioned earlier any permutation of (a) and (b) will also give us a new schedule. For both (a) and (b) the number of permutations are
=
So the total number of schedules is =
.
Solution 2
Label the players of the first team ,
, and
, and those of the second team,
,
, and
. We can start by assigning an opponent to person
for all
games. Since
has to play each of
,
, and
twice, there are
ways to do this. We can assume that the opponents for
in the
rounds are
,
,
,
,
,
and multiply by
afterwards.
Notice that for every valid assignment of the opponents of and
, there is only
valid assignment of opponents for
. More specifically, the opponents for
are the leftover opponents after the opponents for
and
are chosen in each round. Therefore, all we have to do is assign the opponents for
. This is the same as finding the number of permutations of
,
, and
that do not have a
in the first two spots, an
in the next two spots, and a
in the final two spots.
We can use casework to find this by using the fact that after we put down the 's and
's first there is
way to put down the
's (the two remaining spots).
If 's are put in the middle
spots, then there is
way to assign spots to
, namely the last
spots. (If one of the last two spots are left empty, there will have to be a
there, which which not valid).
If 's are put in the last
spots, then there is
way to assign spots to
.
Finally, if one was put in on of the middle two spots and one
was put in one of the last two spots, there are
ways to assign spots to
and
ways to assign spots to
(one of the first two spots and the remaining spot in the last
).
There are ways to assign opponents to
and
ways to order the games.
Solution 3(similar to 2)
We only need to worry about what the other member of the pair would be for our own competitors. If our competitors are and the others are
, then we only need to arrange the numbers
such that we don't have the same number in the same spots. For our first row of numbers, or our first player, they can be arranged in any of
ways as nothing has happened yet. This contributes
to our final product. Once we determine our second row, or player, the third will immediately follow thus we only need to figure out the second one. WLOG, our first arrangement was
. For our
's, we can put them anywhere that is not in our first
spots. If we put both of them in the same "region", like the
's or the
's, then it will give a different case than if they were in different regions.
Case 1 : They are in the same region
There are ways that the
's can be in the same region. If, let's say its in the
's region, we try placing the
, we can only place them in the
region in order to fit the
's. Therefore, this entire case produces
arrangements. \\
Case 2: They are in a different region
There are ways that the
's can be in different regions. For the
's and
's, they will have to go in each other's region with
and the other can be arranged in
ways in the
region giving a total of
for this case.\\
Finally,
~Vedoral
Graph Theory Insight
This is a bipartite graph perfect matching problem. This bipartite graph has perfect matchings. Arranging these 6 perfect matchings in 6 rounds gives
ways of scheduling, as Solution 1 states. But Solution 1 did not write about why there are 90 ways to make the schedule that two rounds are repeated, and why 90 is added twice.
Let's call round 1 and 2
, round 3 and 4
, round 5 and 6
.
Round 1---->![]()
![]()
![]()
Round 2---->
![]()
![]()
![]()
Round 3---->
![]()
![]()
![]()
Round 4---->
![]()
![]()
![]()
Round 5---->
![]()
![]()
![]()
Round 6---->
![]()
![]()
![]()
![]()
And now the schedule is simply the permutation of , which is:
=
So now we have to figure out how many
s there are. We first put in the
,
, and
:
---->
![]()
---->
![]()
---->
![]()
Then there are 2 ways to put in 's match in
:
a)
---->
![]()
![]()
---->
![]()
---->
![]()
b)
---->
![]()
![]()
---->
![]()
---->
![]()
Now we fill in 's match for
and
.
Notice how that for each a) and b) there is only one way to fill in
's match for
and
. For a), if we put
in
, then
must be in
. But
is already in
, which makes it impossible. For b), if we put
in
,
must be in
. Then,
has to be in both
and
which is impossible. So there is only one way to fill in
's match for
and
.
a)
---->
![]()
![]()
---->
![]()
![]()
---->
![]()
![]()
b)
---->
![]()
![]()
---->
![]()
![]()
---->
![]()
![]()
Now we fill in 's matchs:
a)
---->
![]()
![]()
![]()
---->
![]()
![]()
![]()
---->
![]()
![]()
![]()
b)
---->
![]()
![]()
![]()
---->
![]()
![]()
![]()
---->
![]()
![]()
![]()
There are 2 s, therefore 90 must be added twice.
Video Solution by Richard Rusczyk
https://artofproblemsolving.com/videos/amc/2013amc10a/358
See Also
2013 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.