Difference between revisions of "2022 AMC 12B Problems/Problem 17"
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==Problem== | ==Problem== | ||
− | How many <math>4 \times 4</math> arrays whose entries are <math>0</math>s and <math>1</math>s are there such that the row sums (the sum of the | + | How many <math>4 \times 4</math> arrays whose entries are <math>0</math>s and <math>1</math>s are there such that the row sums (the sum of the entries in each row) are <math>1, 2, 3,</math> and <math>4,</math> in some order, and the column sums (the sum of the entries in each column) are also <math>1, 2, 3,</math> and <math>4,</math> in some order? For example, the array |
− | entries in each row) are 1, 2, 3, and 4, in some order, and the column sums (the sum of the entries in | + | <cmath>\left[ |
− | each column) are also 1, 2, 3, and 4, in some order? For example, the array | ||
− | <cmath> | ||
− | |||
− | \left[ | ||
\begin{array}{cccc} | \begin{array}{cccc} | ||
1 & 1 & 1 & 0 \\ | 1 & 1 & 1 & 0 \\ | ||
Line 13: | Line 9: | ||
0 & 1 & 0 & 0 \\ | 0 & 1 & 0 & 0 \\ | ||
\end{array} | \end{array} | ||
− | \right] | + | \right]</cmath> |
− | |||
− | </cmath> | ||
− | |||
satisfies the condition. | satisfies the condition. | ||
<math>\textbf{(A) }144 \qquad \textbf{(B) }240 \qquad \textbf{(C) }336 \qquad \textbf{(D) }576 \qquad \textbf{(E) }624</math> | <math>\textbf{(A) }144 \qquad \textbf{(B) }240 \qquad \textbf{(C) }336 \qquad \textbf{(D) }576 \qquad \textbf{(E) }624</math> | ||
− | ==Solution (Linear | + | ==Solution 1 (One-to-One Correspondence)== |
+ | |||
+ | Note that the arrays and the sum configurations have one-to-one correspondence. Furthermore, the row sum configuration and the column sum configuration are independent of each other. Therefore, the answer is <math>(4!)^2=\boxed{\textbf{(D) }576}.</math> | ||
+ | |||
+ | ~bad_at_mathcounts ~MRENTHUSIASM | ||
+ | |||
+ | ==Solution 2 (Linear Transformation and Permutation)== | ||
In this problem, we call a matrix that satisfies all constraints given in the problem a feasible matrix. | In this problem, we call a matrix that satisfies all constraints given in the problem a feasible matrix. | ||
Line 30: | Line 29: | ||
Second, we observe that there is a unique benchmark matrix, as shown below: | Second, we observe that there is a unique benchmark matrix, as shown below: | ||
− | <cmath> | + | <cmath>\left[ |
− | |||
− | \left[ | ||
\begin{array}{cccc} | \begin{array}{cccc} | ||
0 & 0 & 0 & 1 \\ | 0 & 0 & 0 & 1 \\ | ||
Line 39: | Line 36: | ||
1 & 1 & 1 & 1 \\ | 1 & 1 & 1 & 1 \\ | ||
\end{array} | \end{array} | ||
− | \right] | + | \right]</cmath> |
− | |||
− | </cmath> | ||
− | |||
With above observations, we now count the number of feasible matrixes. | With above observations, we now count the number of feasible matrixes. | ||
We construct a feasible matrix in the following steps. | We construct a feasible matrix in the following steps. | ||
Line 55: | Line 49: | ||
Following from the rule of product, the total number of feasible matrixes is <math>4! \cdot 4! = | Following from the rule of product, the total number of feasible matrixes is <math>4! \cdot 4! = | ||
− | \boxed{\textbf{(D) 576 | + | \boxed{\textbf{(D) }576}</math>. |
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
− | == | + | ==Solution 3 (Multiplication Principle)== |
+ | |||
+ | Of the sixteen entries in one such array, there are six <math>0</math>s and ten <math>1</math>s. The rows and the columns each have zero, one, two, and three <math>0</math>s, in some order. Once we decide the positions of the <math>0</math>s, we form one such array. | ||
+ | |||
+ | We perform the following steps: | ||
+ | <ol style="margin-left: 1.5em;"> | ||
+ | <li>There are <math>\binom41=4</math> ways to choose the row with three <math>0</math>s. After that, there are <math>\binom43=4</math> ways to choose the positions of the three <math>0</math>s.</li><p> | ||
+ | <li>There are <math>\binom31=3</math> ways to choose the column with three <math>0</math>s. After that, there are <math>\binom32=3</math> ways to choose the positions of the two additional <math>0</math>s.</li><p> | ||
+ | <li>There are <math>\binom41=4</math> ways to choose the position of the last <math>0:</math> It is the intersection of the column of one <math>0</math> in Step 1 and the row of one <math>0</math> in Step 2.</li><p> | ||
+ | </ol> | ||
+ | Together, the answer is <math>4\cdot4\cdot3\cdot3\cdot4=\boxed{\textbf{(D) }576}.</math> | ||
+ | |||
+ | ~MRENTHUSIASM | ||
+ | |||
+ | ==Solution 4 (Multiplication Principle)== | ||
+ | |||
Since exactly <math>1</math> row sum is <math>4</math> and exactly <math>1</math> column sum is <math>4</math>, there is a unique entry in the array such that it, and every other entry in the same row or column, is a <math>1.</math> Since there are <math>16</math> total entries in the array, there are <math>16</math> ways to choose the entry with only <math>1</math>s in its row and column. | Since exactly <math>1</math> row sum is <math>4</math> and exactly <math>1</math> column sum is <math>4</math>, there is a unique entry in the array such that it, and every other entry in the same row or column, is a <math>1.</math> Since there are <math>16</math> total entries in the array, there are <math>16</math> ways to choose the entry with only <math>1</math>s in its row and column. | ||
− | + | Without loss of generality, let that entry be in the top-left corner of the square. Note that there is already <math>1</math> entry numbered <math>1</math> in each unfilled row, and <math>1</math> entry numbered <math>1</math> in each unfilled column. Since exactly <math>1</math> row sum is <math>1</math> and exactly <math>1</math> column sum is <math>1</math>, there is a unique entry in the <math>3\times3</math> array of the empty squares such that it, and every other entry in the same row or column in the <math>3\times3</math> array is a <math>0.</math> Using a process similar to what we used in the first paragraph, we can see that there are <math>9</math> ways to choose the entry with only <math>0</math> in its row and column (in the <math>3\times3</math> array). | |
− | + | Without loss of generality, let that entry be in the bottom-right corner of the square. Then, the remaining empty squares are the <math>4</math> center squares. Of these, one of the columns of the empty <math>2\times2</math> array will have two <math>1</math>s and the other column will have one <math>1.</math> That happens if and only if exactly <math>1</math> of the remaining squares is filled with a <math>0</math>, and there are <math>4</math> ways to choose that square. Filling that square with a <math>0</math> and the other <math>3</math> squares with <math>1</math>s completes the grid. | |
− | All in all, there are <math>4 | + | All in all, there are <math>4\cdot9\cdot16=\boxed{\textbf{(D) }576}</math> ways to complete the grid. |
− | pianoboy | + | ~pianoboy |
− | ~mathboy100 (minor LaTeX fix) | + | |
+ | ~mathboy100 (minor <math>\LaTeX</math> fix) | ||
+ | |||
+ | ==Solution 5 (Meta Solving / Conjectures)== | ||
+ | Note that swapping any two rows or any two columns or both from the given example array, leads to a new array that satisfies the condition. There are <math>4</math> rows, and you choose <math>2</math> to swap, so <math>\frac{4!}{2!2!} = 6</math>; likewise for columns. Therefore, <math>6\cdot6 = 36</math>. Clearly, the final answer must be divisible by <math>36</math>, so we can eliminate <math>\textbf{(B)}</math>, <math>\textbf{(C)}</math> and <math>\textbf{(E)}</math>. | ||
+ | |||
+ | Now, you are in exam mode with not much time left. You see that <math>144 = 4\cdot36</math> while <math>576 = 16\cdot36</math>. There are <math>16</math> elements in the array (<math>4</math> rows and <math>4</math> columns), so you go with <math>\boxed{\textbf{(D) }576}</math>, and this is indeed the correct answer! | ||
+ | |||
+ | ~alphamugamma | ||
+ | |||
+ | |||
+ | ==Video Solution (Quick)== | ||
+ | https://youtu.be/NIuo5i3nFzo | ||
+ | |||
+ | <i>~Education, the Study of Everything</i> | ||
==Video Solution== | ==Video Solution== | ||
Line 76: | Line 99: | ||
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/JVDlHCSPF6k | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2022|ab=B|num-b=16|num-a=18}} | {{AMC12 box|year=2022|ab=B|num-b=16|num-a=18}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 00:19, 10 July 2023
Contents
Problem
How many arrays whose entries are
s and
s are there such that the row sums (the sum of the entries in each row) are
and
in some order, and the column sums (the sum of the entries in each column) are also
and
in some order? For example, the array
satisfies the condition.
Solution 1 (One-to-One Correspondence)
Note that the arrays and the sum configurations have one-to-one correspondence. Furthermore, the row sum configuration and the column sum configuration are independent of each other. Therefore, the answer is
~bad_at_mathcounts ~MRENTHUSIASM
Solution 2 (Linear Transformation and Permutation)
In this problem, we call a matrix that satisfies all constraints given in the problem a feasible matrix.
First, we observe that if a matrix is feasible, and we swap two rows or two columns to get a new matrix, then this new matrix is still feasible.
Therefore, any feasible matrix can be obtained through a sequence of such swapping operations from a feasible matrix where for all , the sum of entries in row
is
and the sum of entries in column
is
, hereafter called as a benchmark matrix.
Second, we observe that there is a unique benchmark matrix, as shown below:
With above observations, we now count the number of feasible matrixes.
We construct a feasible matrix in the following steps.
Step 1: We make a permutation of rows of the benchmark matrix.
The number of ways is .
Step 2: We make a permutation of columns of the matrix obtained after Step 1.
The number of ways is .
Following from the rule of product, the total number of feasible matrixes is .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 3 (Multiplication Principle)
Of the sixteen entries in one such array, there are six s and ten
s. The rows and the columns each have zero, one, two, and three
s, in some order. Once we decide the positions of the
s, we form one such array.
We perform the following steps:
- There are
ways to choose the row with three
s. After that, there are
ways to choose the positions of the three
s.
- There are
ways to choose the column with three
s. After that, there are
ways to choose the positions of the two additional
s.
- There are
ways to choose the position of the last
It is the intersection of the column of one
in Step 1 and the row of one
in Step 2.
Together, the answer is
~MRENTHUSIASM
Solution 4 (Multiplication Principle)
Since exactly row sum is
and exactly
column sum is
, there is a unique entry in the array such that it, and every other entry in the same row or column, is a
Since there are
total entries in the array, there are
ways to choose the entry with only
s in its row and column.
Without loss of generality, let that entry be in the top-left corner of the square. Note that there is already entry numbered
in each unfilled row, and
entry numbered
in each unfilled column. Since exactly
row sum is
and exactly
column sum is
, there is a unique entry in the
array of the empty squares such that it, and every other entry in the same row or column in the
array is a
Using a process similar to what we used in the first paragraph, we can see that there are
ways to choose the entry with only
in its row and column (in the
array).
Without loss of generality, let that entry be in the bottom-right corner of the square. Then, the remaining empty squares are the center squares. Of these, one of the columns of the empty
array will have two
s and the other column will have one
That happens if and only if exactly
of the remaining squares is filled with a
, and there are
ways to choose that square. Filling that square with a
and the other
squares with
s completes the grid.
All in all, there are ways to complete the grid.
~pianoboy
~mathboy100 (minor fix)
Solution 5 (Meta Solving / Conjectures)
Note that swapping any two rows or any two columns or both from the given example array, leads to a new array that satisfies the condition. There are rows, and you choose
to swap, so
; likewise for columns. Therefore,
. Clearly, the final answer must be divisible by
, so we can eliminate
,
and
.
Now, you are in exam mode with not much time left. You see that while
. There are
elements in the array (
rows and
columns), so you go with
, and this is indeed the correct answer!
~alphamugamma
Video Solution (Quick)
~Education, the Study of Everything
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
See Also
2022 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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.