Difference between revisions of "2017 AMC 12B Problems/Problem 22"
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− | It amounts to filling in a | + | It amounts to filling in a <math>4 \times 4</math> matrix. Columns <math>C_1 - C_4</math> are the random draws each round; rows <math>R_A - R_D</math> are the coin changes of each player. Also, let <math>\%R_A</math> be the number of nonzero elements in <math>R_A</math>. |
− | WLOG, let | + | WLOG, let <math>C_1 = \begin{pmatrix} 1\\-1\\0\\0\end{pmatrix}</math>. Parity demands that <math>\%R_A</math> and <math>\%R_B</math> must equal <math>2</math> or <math>4</math>. |
− | + | Case 1: <math>\%R_A = 4</math> and <math>\%R_B = 4</math>. There are <math>\binom{3}{2}=3</math> ways to place <math>2</math> <math>-1</math>'s in <math>R_A</math>, so there are <math>3</math> ways. | |
− | Case 2 | + | Case 2: <math>\%R_A = 2</math> and <math>\%R_B=4</math>. There are <math>3</math> ways to place the <math>-1</math> in <math>R_A</math>, <math>2</math> ways to place the remaining <math>-1</math> in <math>R_B</math> (just don't put it under the <math>-1</math> on top of it!), and <math>2</math> 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 <math>\%R_A = 4</math>, <math>\%R_B = 2</math> for a total of <math>24</math> ways. |
− | Case 3 | + | Case 3: <math>\%R_A=\%R_B=2</math>. 3 Choices for -1in RA. Then (1)1 in RB goes directly under 1 in RA. Then 2 choices to assign remaining two pairs of 1, -1in RC and RD. Or (2) 1 in RB goes not directly under 1 in RA: 2 choices here. Then 2 choices to finish. Hence 3(2+2×2)=18. |
In sum, 3+24+18=45 cases. Probability= 45÷(12^3)=5/192. | In sum, 3+24+18=45 cases. Probability= 45÷(12^3)=5/192. |
Revision as of 15:00, 10 September 2017
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: . 3 Choices for -1in RA. Then (1)1 in RB goes directly under 1 in RA. Then 2 choices to assign remaining two pairs of 1, -1in RC and RD. Or (2) 1 in RB goes not directly under 1 in RA: 2 choices here. Then 2 choices to finish. Hence 3(2+2×2)=18.
In sum, 3+24+18=45 cases. Probability= 45÷(12^3)=5/192.
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 |
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