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Difference between revisions of "2002 AIME II Problems/Problem 9"

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== Problem ==
 
== Problem ==
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Let <math>\mathcal{S}</math> be the [[set]] <math>\lbrace1,2,3,\ldots,10\rbrace</math> Let <math>n</math> be the number of sets of two non-empty disjoint subsets of <math>\mathcal{S}</math>. (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when <math>n</math> is divided by <math>1000</math>.
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Let <math>\mathcal{S}</math> be the [[set]] <math>\lbrace1,2,3,\ldots,10\rbrace</math>. Let <math>n</math> be the number of sets of two non-empty disjoint subsets of <math>\mathcal{S}</math>. (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when <math>n</math> is divided by <math>1000</math>.
  
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== Solution 1 ==
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== Solution 1 (combinatorial) ==
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Let the two disjoint subsets be <math>A</math> and <math>B</math>, and let <math>C = \mathcal{S}-(A+B)</math>. For each <math>i \in \mathcal{S}</math>, either <math>i \in A</math>, <math>i \in B</math>, or <math>i \in C</math>. So there are <math>3^{10}</math> ways to organize the elements of <math>S</math> into disjoint <math>A</math>, <math>B</math>, and <math>C</math>.  
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Let the two disjoint subsets be <math>A</math> and <math>B</math>, and define <math>C = \mathcal{S}\setminus\left(A \cup B\right)</math> (so <math>C</math> is disjoint from both <math>A</math> and <math>B</math>, and every element of <math>\mathcal{S}</math>, if not contained in <math>A</math> or <math>B</math>, is necessarily contained in <math>C</math>). This means every element <math>s \in \mathcal{S}</math> satisfies exactly one of <math>s \in A</math>, <math>s \in B</math>, or <math>s \in C</math>, i.e. for each of the <math>10</math> elements, we have to independently choose <math>1</math> of these <math>3</math> subsets into which to place it. Hence there are a total of <math>3^{10}</math> ways to partition <math>\mathcal{S}</math> into disjoint subsets <math>A</math>, <math>B</math>, and <math>C</math>.  
  
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However, there are <math>2^{10}</math> ways to organize the elements of <math>\mathcal{S}</math> such that <math>A = \emptyset</math> and <math>\mathcal{S} = B+C</math>, and there are <math>2^{10}</math> ways to organize the elements of <math>\mathcal{S}</math> such that <math>B = \emptyset</math> and <math>\mathcal{S} = A+C</math>.  
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However, we must exclude those partitions in which <math>A</math> or <math>B</math> (or both) is empty. If <math>A = \emptyset</math>, and thus <math>\mathcal{S} = B \cup C</math>, then we have to place each of the <math>10</math> elements of <math>\mathcal{S}</math> into either <math>B</math> or <math>C</math>, giving <math>2^{10}</math> such partitions (similarly to above). By symmetry, the case where <math>B = \emptyset</math> and <math>\mathcal{S} = A \cup C</math> also gives <math>2^{10}</math> partitions, so we need to subtract <math>2^{10}</math> twice from <math>3^{10}</math>. But then the partition <math>A = B = \emptyset, \mathcal{S} = C</math>, which falls within ''both'' of the above <math>2</math> cases, would be double-counted in the subtraction, so we must add it back.  
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But, the combination such that <math>A = B = \emptyset</math> and <math>\mathcal{S} = C</math> is counted twice.  
 
  
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Thus, there are <math>3^{10}-2\cdot2^{10}+1</math> ordered pairs of sets <math>(A,B)</math>. But since the question asks for the number of unordered sets <math>\{ A,B \}</math>, <math>n = \frac{1}{2}(3^{10}-2\cdot2^{10}+1) = 28501 \equiv \boxed{501} \pmod{1000}</math>.   
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Accordingly, there are <math>\left(3^{10}-2\cdot2^{10}+1\right)</math> possible ordered pairs <math>(A,B)</math> (or equivalently, partitions <math>A,B,C</math>). Each pair has <math>2</math> possible orders (as <math>A</math> and <math>B</math>, being disjoint, obviously cannot be the same), so the above expression counts each ''unordered'' set <math>\left\{A,B\right\}</math> exactly twice. Thus, finally, we deduce that <math>n = \frac{1}{2}\left(3^{10}-2\cdot2^{10}+1\right) = 28501 \equiv \boxed{501} \pmod{1000}</math>.   
  
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== Solution 2 ==
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== Solution 2 (computational/'bash') ==
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Let <math>A</math> and <math>B</math> be the disjoint subsets. If <math>A</math> has <math>n</math> elements, then the number of elements of <math>B</math> can be any positive integer number less than or equal to <math>10-n</math>. So <math>2n=\binom{10}{1} \cdot \left(\binom{9}{1}+\binom{9}{2}+\dots +\binom{9}{9}\right)+\binom{10}{2} \cdot \left(\binom{8}{1}+\binom{8}{2}+\dots +\binom{8}{8}\right)+\dots +\binom{10}{9} \cdot \binom{1}{1}=</math>
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As in Solution 1, let the two disjoint subsets be <math>A</math> and <math>B</math>. We observe that if <math>A</math> has <math>p</math> elements, which can be chosen in <math>\binom{10}{p}</math> ways, then there are <math>(10-p)</math> remaining elements of <math>\mathcal{S}</math> from which to choose the elements of <math>B</math>.
  
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<math>=\binom{10}{1} \cdot \sum_{n=1}^9 \binom{9}{n}+\binom{10}{2} \cdot \sum_{n=1}^8 \binom{8}{n}+\dots + \binom{10}{9} \cdot \binom{1}{1}=</math>
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Therefore, letting <math>B</math> have <math>q</math> elements, we must have <math>1 \leq q \leq 10-p</math>, and similarly to above, these <math>q</math> elements can be chosen in <math>\binom{10-p}{q}</math> ways. Moreover, since <math>A</math> and <math>B</math> must both be non-empty, we require <math>p \geq 1</math> and <math>10-p \geq 1</math>, i.e. <math>1 \leq p \leq 9</math>. It follows that for each fixed value of <math>p</math>, the number of possible ordered pairs <math>(A,B)</math> is <cmath>\begin{align*}\binom{10}{p}\sum_{q=1}^{10-p}\binom{10-p}{q} &= \binom{10}{p}\sum_{q=1}^{10-p}\binom{10-p}{q}1^q 1^{(10-p)-q} \\ &= \binom{10}{p}\left(\sum_{q=0}^{10-p}\binom{10-p}{q}1^q 1^{(10-p)-q} - \binom{10-p}{0}\cdot 1^0 \cdot 1^{(10-p)-0}\right) \\ &= \binom{10}{p}\left(\left(1+1\right)^{10-p} - 1 \cdot 1 \cdot 1\right) \qquad \text{(by the binomial theorem)} \\ &= \binom{10}{p}\left(2^{10-p}-1\right),\end{align*}</cmath> and summing this expression over <math>1 \leq p \leq 9</math> will therefore give the total number of such ordered pairs.
  
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<math>=\binom{10}{1} \cdot \left(2^9-1\right)+\binom{10}{2} \cdot \left(2^8-1\right)+\dots+\binom{10}{9} \cdot \left(2-1\right)=</math>
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Just like in Solution 1, counting ordered pairs <math>(A,B)</math> is equivalent to counting each unordered set <math>\left\{A,B\right\}</math> exactly twice, so we deduce <cmath>\begin{align*}2n &= \sum_{p=1}^{9}\binom{10}{p}\left(2^{10-p}-1\right) \\ &= \sum_{p=1}^{9}\binom{10}{p}1^p 2^{10-p} - \sum_{p=1}^{9}\binom{10}{p}1^p 1^{10-p} \\ &= \left(\sum_{p=0}^{10}\binom{10}{p}1^p 2^{10-p} - \binom{10}{0} \cdot 1^0 \cdot 2^{10-0} - \binom{10}{10} \cdot 1^{10} \cdot 2^{10-10}\right) \\ &\phantom{=} -\left(\sum_{p=0}^{10}\binom{10}{p}1^p 1^{10-p} - \binom{10}{0} \cdot 1^0 \cdot 1^{10-0} - \binom{10}{10} \cdot 1^{10} \cdot 1^{10-10}\right) \\ &= \left(\left(1+2\right)^{10} - 1 \cdot 1 \cdot 2^{10} - 1 \cdot 1 \cdot 1\right) - \left(\left(1+1\right)^{10} - 1 \cdot 1 \cdot 1 - 1 \cdot 1 \cdot 1\right) \\ &\phantom{=} \text{(again by the binomial theorem)} \\ &= 3^{10}-2^{10}-1-2^{10}+1+1 \\ &= 3^{10} - 2 \cdot 2^{10} + 1.\end{align*}</cmath> Hence, as in Solution 1, <math>n = \frac{1}{2}\left(3^{10}-2\cdot2^{10}+1\right) = 28501 \equiv \boxed{501} \pmod{1000}</math>.
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<math>=\sum_{n=0}^{10} \binom{10}{n} 2^{10-n} 1^n - \binom{10}{0} \cdot 2^{10} - \binom{10}{10}-\left(\sum_{n=0}^{10} \binom{10}{n} - \binom{10}{0} - \binom{10}{10} \right) =</math>
 
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<math>=(2+1)^{10}-2^{10}-1-(1+1)^{10}+1+1=3^{10}-2^{11}+1=57002</math>
 
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Then <math>n=\frac{57002}{2}=28501\equiv \boxed{501} \pmod{1000}</math>
 
  
 
== See also ==
 
== See also ==

Latest revision as of 02:53, 17 June 2025

Problem

Let $\mathcal{S}$ be the set $\lbrace1,2,3,\ldots,10\rbrace$. Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}$. (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when $n$ is divided by $1000$.

Solution 1 (combinatorial)

Let the two disjoint subsets be $A$ and $B$, and define $C = \mathcal{S}\setminus\left(A \cup B\right)$ (so $C$ is disjoint from both $A$ and $B$, and every element of $\mathcal{S}$, if not contained in $A$ or $B$, is necessarily contained in $C$). This means every element $s \in \mathcal{S}$ satisfies exactly one of $s \in A$, $s \in B$, or $s \in C$, i.e. for each of the $10$ elements, we have to independently choose $1$ of these $3$ subsets into which to place it. Hence there are a total of $3^{10}$ ways to partition $\mathcal{S}$ into disjoint subsets $A$, $B$, and $C$.

However, we must exclude those partitions in which $A$ or $B$ (or both) is empty. If $A = \emptyset$, and thus $\mathcal{S} = B \cup C$, then we have to place each of the $10$ elements of $\mathcal{S}$ into either $B$ or $C$, giving $2^{10}$ such partitions (similarly to above). By symmetry, the case where $B = \emptyset$ and $\mathcal{S} = A \cup C$ also gives $2^{10}$ partitions, so we need to subtract $2^{10}$ twice from $3^{10}$. But then the partition $A = B = \emptyset, \mathcal{S} = C$, which falls within both of the above $2$ cases, would be double-counted in the subtraction, so we must add it back.

Accordingly, there are $\left(3^{10}-2\cdot2^{10}+1\right)$ possible ordered pairs $(A,B)$ (or equivalently, partitions $A,B,C$). Each pair has $2$ possible orders (as $A$ and $B$, being disjoint, obviously cannot be the same), so the above expression counts each unordered set $\left\{A,B\right\}$ exactly twice. Thus, finally, we deduce that $n = \frac{1}{2}\left(3^{10}-2\cdot2^{10}+1\right) = 28501 \equiv \boxed{501} \pmod{1000}$.

Solution 2 (computational/'bash')

As in Solution 1, let the two disjoint subsets be $A$ and $B$. We observe that if $A$ has $p$ elements, which can be chosen in $\binom{10}{p}$ ways, then there are $(10-p)$ remaining elements of $\mathcal{S}$ from which to choose the elements of $B$.

Therefore, letting $B$ have $q$ elements, we must have $1 \leq q \leq 10-p$, and similarly to above, these $q$ elements can be chosen in $\binom{10-p}{q}$ ways. Moreover, since $A$ and $B$ must both be non-empty, we require $p \geq 1$ and $10-p \geq 1$, i.e. $1 \leq p \leq 9$. It follows that for each fixed value of $p$, the number of possible ordered pairs $(A,B)$ is \begin{align*}\binom{10}{p}\sum_{q=1}^{10-p}\binom{10-p}{q} &= \binom{10}{p}\sum_{q=1}^{10-p}\binom{10-p}{q}1^q 1^{(10-p)-q} \\ &= \binom{10}{p}\left(\sum_{q=0}^{10-p}\binom{10-p}{q}1^q 1^{(10-p)-q} - \binom{10-p}{0}\cdot 1^0 \cdot 1^{(10-p)-0}\right) \\ &= \binom{10}{p}\left(\left(1+1\right)^{10-p} - 1 \cdot 1 \cdot 1\right) \qquad \text{(by the binomial theorem)} \\ &= \binom{10}{p}\left(2^{10-p}-1\right),\end{align*} and summing this expression over $1 \leq p \leq 9$ will therefore give the total number of such ordered pairs.

Just like in Solution 1, counting ordered pairs $(A,B)$ is equivalent to counting each unordered set $\left\{A,B\right\}$ exactly twice, so we deduce \begin{align*}2n &= \sum_{p=1}^{9}\binom{10}{p}\left(2^{10-p}-1\right) \\ &= \sum_{p=1}^{9}\binom{10}{p}1^p 2^{10-p} - \sum_{p=1}^{9}\binom{10}{p}1^p 1^{10-p} \\ &= \left(\sum_{p=0}^{10}\binom{10}{p}1^p 2^{10-p} - \binom{10}{0} \cdot 1^0 \cdot 2^{10-0} - \binom{10}{10} \cdot 1^{10} \cdot 2^{10-10}\right) \\ &\phantom{=} -\left(\sum_{p=0}^{10}\binom{10}{p}1^p 1^{10-p} - \binom{10}{0} \cdot 1^0 \cdot 1^{10-0} - \binom{10}{10} \cdot 1^{10} \cdot 1^{10-10}\right) \\ &= \left(\left(1+2\right)^{10} - 1 \cdot 1 \cdot 2^{10} - 1 \cdot 1 \cdot 1\right) - \left(\left(1+1\right)^{10} - 1 \cdot 1 \cdot 1 - 1 \cdot 1 \cdot 1\right) \\ &\phantom{=} \text{(again by the binomial theorem)} \\ &= 3^{10}-2^{10}-1-2^{10}+1+1 \\ &= 3^{10} - 2 \cdot 2^{10} + 1.\end{align*} Hence, as in Solution 1, $n = \frac{1}{2}\left(3^{10}-2\cdot2^{10}+1\right) = 28501 \equiv \boxed{501} \pmod{1000}$.

See also

2002 AIME II (Problems • Answer Key • Resources)
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Problem 10
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