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

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So the answer is <math>\left(\sum_{n=1}^{11} \dbinom{10}{n-1}\right) - \dbinom{10}{5}=2^{10}-252= \boxed{772}</math>.
 
So the answer is <math>\left(\sum_{n=1}^{11} \dbinom{10}{n-1}\right) - \dbinom{10}{5}=2^{10}-252= \boxed{772}</math>.
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Note: We have <math>\dbinom{10}{n-1}</math>  ways of picking the numbers to be in <math>A</math> because there are <math>n</math> numbers in <math>A</math> and since <math>12-n</math> is already a term in the set we simply have to choose another <math>n-1</math> numbers from the <math>10</math> numbers that are available.
  
 
== Solution 2==
 
== Solution 2==
 
Regardless of the size <math>n</math> of <math>A</math> (ignoring the case when <math>n = 6</math>), <math>n</math> must not be in <math>A</math> and <math>12 - n</math> must be in <math>A</math>.  
 
Regardless of the size <math>n</math> of <math>A</math> (ignoring the case when <math>n = 6</math>), <math>n</math> must not be in <math>A</math> and <math>12 - n</math> must be in <math>A</math>.  
  
There are <math>10</math> remaining elements who’s placements have yet to be determined. Note that the actual value of <math>n</math> does not matter; there is always <math>1</math> necessary element, <math>1</math> forbidden element, and <math>10</math> other elements that need to be distributed. There are <math>2</math> places to put each of these elements, for <math>2^{10}</math> possibilities.
+
There are <math>10</math> remaining elements whose placements have yet to be determined. Note that the actual value of <math>n</math> does not matter; there is always <math>1</math> necessary element, <math>1</math> forbidden element, and <math>10</math> other elements that need to be distributed. There are <math>2</math> places to put each of these elements, for <math>2^{10}</math> possibilities.
  
 
However, there is the edge case of <math>n = 6; 6</math> is forced not the be in either set, so we must subtract the <math>\dbinom{10}{5}</math> cases where <math>A</math> and <math>B</math> have size <math>6</math>.
 
However, there is the edge case of <math>n = 6; 6</math> is forced not the be in either set, so we must subtract the <math>\dbinom{10}{5}</math> cases where <math>A</math> and <math>B</math> have size <math>6</math>.

Latest revision as of 09:36, 6 September 2021

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 $2^{10} - \dbinom{10}{5} = 1024 - 252 = \boxed{772}$

See also

2010 AIME II (ProblemsAnswer KeyResources)
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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