Difference between revisions of "2022 AMC 10B Problems/Problem 18"
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There are <math>\binom{3}{2}</math> ways to choose two equations from <math>\{5, 6, 7\}</math>. WLOG let Equations 5 and 6 be in our system: <math>x + y = 0</math> and <math>x + z = 0</math>. Any equation from <math>\{2, 3, 4\}</math> will force <math>x = y = z = 0</math>. There are <math>3!</math> ways to arrange the equations. The number of options that force <math>x = y = z = 0</math> is <math>\binom{3}{2} \cdot \binom{3}{1} \cdot 3! = 54</math>. | There are <math>\binom{3}{2}</math> ways to choose two equations from <math>\{5, 6, 7\}</math>. WLOG let Equations 5 and 6 be in our system: <math>x + y = 0</math> and <math>x + z = 0</math>. Any equation from <math>\{2, 3, 4\}</math> will force <math>x = y = z = 0</math>. There are <math>3!</math> ways to arrange the equations. The number of options that force <math>x = y = z = 0</math> is <math>\binom{3}{2} \cdot \binom{3}{1} \cdot 3! = 54</math>. | ||
− | Case <math>2^c</math>: One equation from {5, 6, 7} is present. Two equations from <math>\{2, 3, 4\}</math> are present. | + | Case <math>2^c</math>: One equation from <math>\{5, 6, 7\}</math> is present. Two equations from <math>\{2, 3, 4\}</math> are present. |
− | There are <math>\binom{3}{1}</math> ways to choose one equation from <math>\{5, 6, 7\}</math>. WLOG let Equation 5 (<math>x + y = 0</math>) be present. One of the two equations from <math>\{2, 3, 4\}</math> must be Equation 4, <math>z = 0</math>, since it is the only equation that restricts <math>z</math>. The last equation can be either | + | There are <math>\binom{3}{1}</math> ways to choose one equation from <math>\{5, 6, 7\}</math>. WLOG let Equation 5 (<math>x + y = 0</math>) be present. One of the two equations from <math>\{2, 3, 4\}</math> must be Equation 4, <math>z = 0</math>, since it is the only equation that restricts <math>z</math>. The last equation can be either 2 or 3. There are <math>3!</math> ways to arrange the equations. The number of options that force <math>x = y = z = 0</math> is <math>\binom{3}{1} \cdot \binom{2}{1} \cdot 3! = 36</math>. |
<math>\textbf{Case 3:}</math> Only equations <math>\{2, 3, 4\}</math> are present. | <math>\textbf{Case 3:}</math> Only equations <math>\{2, 3, 4\}</math> are present. |
Revision as of 02:24, 25 November 2022
Contents
Problem
Consider systems of three linear equations with unknowns ,
, and
,
where each of the coefficients is either
or
and the system has a solution other than
.
For example, one such system is
with a nonzero solution of
. How many such systems of equations are there?
(The equations in a system need not be distinct, and two systems containing the same equations in a
different order are considered different.)
Solution 1 (Casework)
Let and
We wish to count the ordered triples of row matrices. We perform casework:
Solution 2 (Complementary Counting)
We will use complementary counting and do casework on the equations.
There are possible equations:
Equation 1:
Equation 2:
Equation 3:
Equation 4:
Equation 5:
Equation 6:
Equation 7:
Equation 8:
We will continue to refer to the equations by their number on this list.
total systems. Note that no two equations by themselves can force
. Therefore no system with Equation 1 or with repeated equations can force
.
Equation 8 (
) is present.
Case : Equation 8, and two equations from
.
There are ways to choose two equations from
and
ways to arrange each case. The number of options that force
is
.
Case : Equation 8, one equation from
, and one equation from
.
There are ways to choose one equation from
. WLOG let us choose Equation 7. Given
and
, we conclude that
. The third equation can be either
or
. There are
ways to arrange each case. The number of options that force
is
.
Case : Equation 8, and two equations from
.
There are ways to choose two equations from
and
ways to arrange each case. Each of these cases forces
.
total options.
Equation 8 is
present, at least one equation from
is present.
Case : Equations
are all present.
There are ways to arrange the three equations.
options.
Case : Two equations from
are present. One equation from
is present.
There are ways to choose two equations from
. WLOG let Equations 5 and 6 be in our system:
and
. Any equation from
will force
. There are
ways to arrange the equations. The number of options that force
is
.
Case : One equation from
is present. Two equations from
are present.
There are ways to choose one equation from
. WLOG let Equation 5 (
) be present. One of the two equations from
must be Equation 4,
, since it is the only equation that restricts
. The last equation can be either 2 or 3. There are
ways to arrange the equations. The number of options that force
is
.
Only equations
are present.
There are ways to arrange the three equations.
options.
We add up the cases: total systems force
. Thus
do not.
~numerophile
Solution 3 (Complementary Counting, Linear Dependence, Vector Analysis)
Denote vector for
.
Thus, we need to count how many vector tuples
are linearly dependent.
We do complementary counting.
First, the total number of vector tuples is
.
Second, we count how many many vector tuples are linearly independent.
To meet this condition, no vector can be a zero vector .
Next, we do the casework analysis.
Case : Three vectors are all on axes.
In this case, the number of is
.
Case : Two vectors are on axes and the third vector is not.
We construct such an instance in the following steps.
Step 1: We determine which two vectors lie on axes.
The number of ways is .
Step 2: For two vectors selected in Step 1, we determine which two axes they lie on.
The number of ways is .
Step 3: For the third unselected vector, we determine its value.
To make three vectors linear independent, the third vector cannot be on the plane formed by the first two vectors.
So the number of ways is .
Following from the rule of product, the number of in this case is
.
Case : One vector is on an axis and the other two are not.
We construct such an instance in the following steps.
Step 1: We determine which vector lies on an axis.
The number of ways is .
Step 2: For the selected vector, we determine which axis it lies on.
The number of ways is .
Step 3: We determine the values of the two unselected vectors.
First, to be linearly independent, these two vectors are distinct.
Second, to be linearly independent, we cannot have one vector and another one that is a diagonal vector on the plane that is perpendicular to the first selected vector.
Thus, the number or ways in this step is .
Following from the rule of product, the number of in this case is
.
Case : No vector is on any axis.
In this case, any three distinct vectors are linearly independent.
So the number of in this case is
.
Putting all cases together, the number of vector tuples that are linearly independent is
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution by OmegaLearn Using Casework
~ pi_is_3.14
See Also
2022 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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 |
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