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Difference between revisions of "2010 AIME II Problems/Problem 8"

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We apply the Principle of Inclusion and Exclusion for casework on the complementary cases. We find the ways where <math>|A|</math> is in <math>A</math>, which violates the first condition. Then we find the ways where the elements <math>|B|</math> and <math>12-|B|</math> are in set <math>B</math>, which violates only the second condition, and not the first. This ensures we do not overcount.
 
We apply the Principle of Inclusion and Exclusion for casework on the complementary cases. We find the ways where <math>|A|</math> is in <math>A</math>, which violates the first condition. Then we find the ways where the elements <math>|B|</math> and <math>12-|B|</math> are in set <math>B</math>, which violates only the second condition, and not the first. This ensures we do not overcount.
  
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Case <math>1</math>: <math>|A|</math> is an element in <math>A</math>
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Case 1: <math>|A|</math> is an element in <math>A</math>
  
 
There are <math>\sum_{i=1}^{11}\dbinom{11}{i-1}</math> = <math>2^{11}-1</math> ways in this case.
 
There are <math>\sum_{i=1}^{11}\dbinom{11}{i-1}</math> = <math>2^{11}-1</math> ways in this case.
  
−
Case <math>2</math>: <math>|B|</math> and <math>12-|B|</math> are in <math>B</math>
+
Case 2: <math>|B|</math> and <math>12-|B|</math> are in <math>B</math>
  
−
We introduce a subcase where <math>|B|</math> is not <math>6</math>, since it would also result in <math>12-6=6</math> for the other element.
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We introduce a subcase where <math>|B|</math> is not 6, since the other element would also be 6.
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There are <math>B-2</math> elements to choose from <math>10</math> elements. Therefore, <math>|B|</math> can be from <math>2</math> to <math>11</math>.
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There are <math>B-2</math> elements to choose from 10 elements. Therefore, <math>|B|</math> can be from 2 to 11.
 
There are <math>\sum_{i=2}^{11}\dbinom{10}{i-2}-\dbinom{10}{4} = 2^{10}-211</math> ways.
 
There are <math>\sum_{i=2}^{11}\dbinom{10}{i-2}-\dbinom{10}{4} = 2^{10}-211</math> ways.
−
The other subcase is when <math>|B|</math> is equal to <math>6</math>. There are <math>\dbinom{11}{5}=462</math> ways.
+
The other subcase is when <math>|B|</math> is equal to 6. There are <math>\dbinom{11}{5}=462</math> ways.
 
Adding the subcases gives us <math>2^{10}+251</math>.  
 
Adding the subcases gives us <math>2^{10}+251</math>.  
  

Latest revision as of 19:57, 20 July 2023

Problem

Let $N$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ that have the following properties:

  • $\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}$,
  • $\mathcal{A} \cap \mathcal{B} = \emptyset$,
  • The number of elements of $\mathcal{A}$ is not an element of $\mathcal{A}$,
  • The number of elements of $\mathcal{B}$ is not an element of $\mathcal{B}$.

Find $N$.

Solution

Let us partition the set $\{1,2,\cdots,12\}$ into $n$ numbers in $A$ and $12-n$ numbers in $B$,

Since $n$ must be in $B$ and $12-n$ must be in $A$ ($n\ne6$, we cannot partition into two sets of 6 because $6$ needs to end up somewhere, $n\ne 0$ or $12$ either).

We have $\dbinom{10}{n-1}$ ways of picking the numbers to be in $A$.

So the answer is $\left(\sum_{n=1}^{11} \dbinom{10}{n-1}\right) - \dbinom{10}{5}=2^{10}-252= \boxed{772}$.

Note: We have $\dbinom{10}{n-1}$ ways of picking the numbers to be in $A$ because there are $n$ numbers in $A$ and since $12-n$ is already a term in the set we simply have to choose another $n-1$ numbers from the $10$ numbers that are available.

Solution 2

Regardless of the size $n$ of $A$ (ignoring the case when $n = 6$), $n$ must not be in $A$ and $12 - n$ must be in $A$.

There are $10$ remaining elements whose placements have yet to be determined. Note that the actual value of $n$ does not matter; there is always $1$ necessary element, $1$ forbidden element, and $10$ other elements that need to be distributed. There are $2$ places to put each of these elements, for $2^{10}$ possibilities.

However, there is the edge case of $n = 6; 6$ is forced not the be in either set, so we must subtract the $\dbinom{10}{5}$ cases where $A$ and $B$ have size $6$.

Thus, our answer is $2^{10} - \dbinom{10}{5} = 1024 - 252 = \boxed{772}$


Solution 3 (PIE and Complementary Counting)

The total number of possible subsets is $\sum_{i=1}^{11}\dbinom{12}{i}$, which is $2^{12}-2$. Note that picking a subset from the set leaves the rest of the set to be in the other subset. We exclude $i=0$ and $i=12$ since they leave a set empty. We proceed with complementary counting and casework:

We apply the Principle of Inclusion and Exclusion for casework on the complementary cases. We find the ways where $|A|$ is in $A$, which violates the first condition. Then we find the ways where the elements $|B|$ and $12-|B|$ are in set $B$, which violates only the second condition, and not the first. This ensures we do not overcount.

Case 1: $|A|$ is an element in $A$

There are $\sum_{i=1}^{11}\dbinom{11}{i-1}$ = $2^{11}-1$ ways in this case.

Case 2: $|B|$ and $12-|B|$ are in $B$

We introduce a subcase where $|B|$ is not 6, since the other element would also be 6. There are $B-2$ elements to choose from 10 elements. Therefore, $|B|$ can be from 2 to 11. There are $\sum_{i=2}^{11}\dbinom{10}{i-2}-\dbinom{10}{4} = 2^{10}-211$ ways. The other subcase is when $|B|$ is equal to 6. There are $\dbinom{11}{5}=462$ ways. Adding the subcases gives us $2^{10}+251$.

Adding this with case one gives us $2^{11}+2^{10}+250$. Subtracting this from $2^{12}-2$ gives $1024-252=\boxed{772}$.

~Magnetoninja

See also

2010 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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