Difference between revisions of "2017 AMC 12B Problems/Problem 22"
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Case 2: <math>3</math> people. In this case, one special person is giving/receiving twice. There are four ways to choose this person, then of the remaining three people we choose two, to be the people giving/receiving to/from the special person. Thus, we have <math>4\cdot\binom{3}{2}\cdot4! = 288</math> ways here. | Case 2: <math>3</math> people. In this case, one special person is giving/receiving twice. There are four ways to choose this person, then of the remaining three people we choose two, to be the people giving/receiving to/from the special person. Thus, we have <math>4\cdot\binom{3}{2}\cdot4! = 288</math> ways here. | ||
− | Case 3: <math>4</math> people. In this case. | + | Case 3: <math>4</math> people. In this case. First note that no person can give/receive twice. If we keep the order of A, B, C, D giving in that order (and permute afterward), then there are three options to choose A's receiver, and three options for B's receiver afterward. Then it is uniquely determined who C and D give to. This gives a total of <math>3\cdot3\cdot4! = 216</math> ways, after permuting. |
So we have a total of <math>36+288+216=540</math> ways to order the four pairs of people. | So we have a total of <math>36+288+216=540</math> ways to order the four pairs of people. |
Revision as of 03:09, 6 February 2019
Problem 22
Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn---one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins?
Solution
It amounts to filling in a matrix. Columns
are the random draws each round; rows
are the coin changes of each player. Also, let
be the number of nonzero elements in
.
WLOG, let . Parity demands that
and
must equal
or
.
Case 1: and
. There are
ways to place
's in
, so there are
ways.
Case 2: and
. There are
ways to place the
in
,
ways to place the remaining
in
(just don't put it under the
on top of it!), and
ways for one of the other two players to draw the green ball. (We know it's green because Bernardo drew the red one.) We can just double to cover the case of
,
for a total of
ways.
Case 3: . There are three ways to place the
in
. Now, there are two cases as to what happens next.
Sub-case 3.1: The in
goes directly under the
in
. There's obviously
way for that to happen. Then, there are
ways to permute the two pairs of
in
and
. (Either the
comes first in
or the
comes first in
.)
Sub-case 3.2: The in
doesn't go directly under the
in
. There are
ways to place the
, and
ways to do the same permutation as in Sub-case 3.1.
Hence, there are
ways for this case.
There's a grand total of ways for this to happen, along with
total cases. The probability we're asking for is thus
Solution 2 (Less Casework)
We will proceed by taking cases based on how many people are taking part in this "transaction." We can have ,
, or
people all giving/receiving coins during the
turns.
Basically, (like the previous solution), we think of this as filling out a
matrix of letters, where a letter on the left column represents this person gave, and a letter on the right column means this person received. We need to make sure that for each person that gave a certain amount, they received in total from other people that same amount, or in other words, we want it such that there are an equal number of A's, B's, C's, and D's on both columns of the matrix.
For example, the matrix below represents: A gives B a coin, then B gives C a coin, C gives D a coin, and finally D gives A a coin, in this order.
Case 1:
people. In this case, we have
ways to choose the two people, and
ways to order them to get a count of
ways.
Case 2: people. In this case, one special person is giving/receiving twice. There are four ways to choose this person, then of the remaining three people we choose two, to be the people giving/receiving to/from the special person. Thus, we have
ways here.
Case 3: people. In this case. First note that no person can give/receive twice. If we keep the order of A, B, C, D giving in that order (and permute afterward), then there are three options to choose A's receiver, and three options for B's receiver afterward. Then it is uniquely determined who C and D give to. This gives a total of
ways, after permuting.
So we have a total of ways to order the four pairs of people.
Now we divide this by the total number of ways:
(four rounds, four ways to choose giver, three to choose receiver each round).
So the answer is
.
~ccx09
Diagram and explanations by no_u
Latex polished by Argonauts16
See Also
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.