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Difference between revisions of "2013 AIME II Problems/Problem 11"

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Let <math>A = \{1, 2, 3, 4, 5, 6, 7\}</math>, and let <math>N</math> be the number of functions <math>f</math> from set <math>A</math> to set <math>A</math> such that <math>f(f(x))</math> is a constant function. Find the remainder when <math>N</math> is divided by <math>1000</math>.
 
Let <math>A = \{1, 2, 3, 4, 5, 6, 7\}</math>, and let <math>N</math> be the number of functions <math>f</math> from set <math>A</math> to set <math>A</math> such that <math>f(f(x))</math> is a constant function. Find the remainder when <math>N</math> is divided by <math>1000</math>.
  
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==Solution==
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==Solution 1==
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Let the range be <math>R=\{f(x)|x \in A\}</math>
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Any such function can be constructed by distributing the elements of <math>A</math> on three tiers.
  
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Let the constant function be <math>f(f(x))=c</math>, clearly, <math>c \in R</math>
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The bottom tier contains the constant value, <math>c=f(f(x))</math> for any <math>x</math>. (Obviously <math>f(c)=c</math>.)
  
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For every <math>a \in R</math>, since <math>f(f(x))=c</math>, we must have <math>f(a)=c</math>.  
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The middle tier contains <math>k</math> elements <math>x\ne c</math> such that <math>f(x)=c</math>, where <math>1\le k\le 6</math>.
  
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Now we enumerate through possible sizes of <math>R</math>. <math>|R|</math> cannot be more than 4 since all the numbers in <math>R</math> must map to <math>c</math> and any number other than <math>c</math> in <math>R</math> must have at least one number mapped to it.  
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The top tier contains <math>6-k</math> elements such that <math>f(x)</math> equals an element on the middle tier.
  
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1. <math>|R|=1</math>
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There are <math>7</math> choices for <math>c</math>. Then for a given <math>k</math>, there are <math>\tbinom6k</math> ways to choose the elements on the middle tier, and then <math>k^{6-k}</math> ways to draw arrows down from elements on the top tier to elements on the middle tier.
  
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We have <math>\binom{7}{1}=7</math> choices for <math>R</math>, <math>\binom{1}{1}=1</math> choices for <math>c</math>, and only 1 way to map the 6 numbers that are not in <math>R</math> (remaining numbers) since the range <math>R</math> is size 1. In total we have <math>7\cdot 1\cdot 1=7</math> choices.  
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Thus <math>N=7\cdot\sum_{k=1}^6\tbinom6k\cdot k^{6-k}=7399</math>, giving the answer <math>\boxed{399}</math>.
  
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2. <math>|R|=2</math>
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==Solution 1 Clarified==
  
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We have <math>\binom{7}{2}=21</math> choices for <math>R</math>, <math>\binom{2}{1}=2</math> choices for <math>c</math>.  
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Define the three layers as [[domain]] <math>x</math>, [[codomain]] <math>f(x)</math>, and codomain <math>f(f(x))</math>. Each one of them is contained in the [[set]] <math>A</math>. We know that <math>f(f(x))</math> is a [[constant function]], or in other words, can only take on one value. So, we can start off by choosing that value <math>c</math> in <math>7</math> ways. So now, we choose the values that can be <math>f(x)</math> for all those values should satisfy <math>f(f(x))=c</math>. Let <math>S</math> be that set of values. First things first, we must have <math>c</math> to be part of <math>S</math>, for the <math>S</math> is part of the domain of <math>x</math>. Since the values in <math>i\in S</math> all satisfy <math>f(i) = c</math>, we have <math>c</math> to be a value that <math>f(x)</math> can be. Now, for the elements other than <math>5</math>:
  
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Assigning the 5 numbers to the two elements in <math>R</math> yields <math>2^5=32</math> choices. However, one of these choices is to assign all 5 remaining numbers to <math>c</math>, resulting in <math>R</math> being only of size 1. Exclude that and we have <math>32-1=31</math> choices.
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If we have <math>k</math> elements other than <math>5</math> that can be part of <math>S</math>, we will have <math>\binom{6}{k}</math> ways to choose those values. There will also be <math>k</math> ways for each of the elements in <math>A</math> other than <math>c</math> and those in set <math>S</math> (for when [[function]] <math>f</math> is applied on those values, we already know it would be <math>c</math>). There are <math>6-k</math> elements in <math>A</math> other than <math>c</math> and those in set <math>S</math>. Thus, there should be <math>k^{6-k}</math> ways to match the domain <math>x</math> to the values of <math>f(x)</math>. Summing up all possible values of <math>k</math> (<math>[1,6]</math>), we have
  
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In total we have <math>21*2*31=1302</math> choices.
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<cmath>\sum_{k=1}^6 \binom{6}{k} k^{6-k} = 6\cdot 1 + 15\cdot 16 + 20\cdot 27 + 15\cdot 16 + 6\cdot 5 + 1 = 1057</cmath>
  
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3. <math>|R|=3</math>
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Multiplying that by the original <math>7</math> for the choice of <math>c</math>, we have <math>7 \cdot 1057 = 7\boxed{399}.</math>
  
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We have <math>\binom{7}{3}=35</math> choices for <math>R</math>, <math>\binom{3}{1}=3</math> choices for <math>c</math>.
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~sml1809
  
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Notice that assigning the remaining numbers is equivalent to distributing 4 numbers into three groups where two of the three groups must receive at least 1 number (since we already have all numbers in <math>R</math> mapped to <math>c</math>, the restriction is not necessary for the <math>c</math> group). Ignoring the restrictions we have <math>3^4</math> ways. Now minus the two cases where one of the two restricted group is left empty. There are 2 ways to choose the left-out group, and <math>2^4</math> ways to distribute numbers for each choice. Finally, by inclusion-exclusion principle, we need to add back the case in which both restricted groups are left empty, which has only <math>1^4=1</math> occurrence. In all, we have <math>3^4-2*2^4+1^4=50</math> ways to assign the remaining numbers.
+
==Video Solution==
 +
https://youtu.be/aaO7abKG0BQ?si=KLfz6oyzVR0d8D13
  
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In total, we have <math>35*3*50=5250</math> choices for this case.  
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~MathProblemSolvingSkills.com
  
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4. <math>|R|=4</math>
 
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We have <math>\binom{7}{4}=35</math> choices for <math>R</math>, <math>\binom{4}{1}=4</math> choices for <math>c</math>.
 
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Now we have 3 "groups" to fill and only 3 remaining numbers. Thus each group must receive exactly 1 number. However, we still have <math>3!=6</math> ways to permute the mappings.
 
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Thus, we have <math>35*4*6=840</math> choices in total.
 
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To summarize, we have <math>7+1302+5250+840=7399</math> different mappings possible. So the answer is <math>\boxed{399}</math>.
 
  
 
==See Also==
 
==See Also==

Revision as of 18:28, 21 September 2024

Problem 11

Let $A = \{1, 2, 3, 4, 5, 6, 7\}$, and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$.

Solution 1

Any such function can be constructed by distributing the elements of $A$ on three tiers.

The bottom tier contains the constant value, $c=f(f(x))$ for any $x$. (Obviously $f(c)=c$.)

The middle tier contains $k$ elements $x\ne c$ such that $f(x)=c$, where $1\le k\le 6$.

The top tier contains $6-k$ elements such that $f(x)$ equals an element on the middle tier.

There are $7$ choices for $c$. Then for a given $k$, there are $\tbinom6k$ ways to choose the elements on the middle tier, and then $k^{6-k}$ ways to draw arrows down from elements on the top tier to elements on the middle tier.

Thus $N=7\cdot\sum_{k=1}^6\tbinom6k\cdot k^{6-k}=7399$, giving the answer $\boxed{399}$.

Solution 1 Clarified

Define the three layers as domain $x$, codomain $f(x)$, and codomain $f(f(x))$. Each one of them is contained in the set $A$. We know that $f(f(x))$ is a constant function, or in other words, can only take on one value. So, we can start off by choosing that value $c$ in $7$ ways. So now, we choose the values that can be $f(x)$ for all those values should satisfy $f(f(x))=c$. Let $S$ be that set of values. First things first, we must have $c$ to be part of $S$, for the $S$ is part of the domain of $x$. Since the values in $i\in S$ all satisfy $f(i) = c$, we have $c$ to be a value that $f(x)$ can be. Now, for the elements other than $5$:

If we have $k$ elements other than $5$ that can be part of $S$, we will have $\binom{6}{k}$ ways to choose those values. There will also be $k$ ways for each of the elements in $A$ other than $c$ and those in set $S$ (for when function $f$ is applied on those values, we already know it would be $c$). There are $6-k$ elements in $A$ other than $c$ and those in set $S$. Thus, there should be $k^{6-k}$ ways to match the domain $x$ to the values of $f(x)$. Summing up all possible values of $k$ ($[1,6]$), we have

\[\sum_{k=1}^6 \binom{6}{k} k^{6-k} = 6\cdot 1 + 15\cdot 16 + 20\cdot 27 + 15\cdot 16 + 6\cdot 5 + 1 = 1057\]

Multiplying that by the original $7$ for the choice of $c$, we have $7 \cdot 1057 = 7\boxed{399}.$

~sml1809

Video Solution

https://youtu.be/aaO7abKG0BQ?si=KLfz6oyzVR0d8D13

~MathProblemSolvingSkills.com


See Also

2013 AIME II (Problems • Answer Key • Resources)
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Problem 10 		Followed by
Problem 12
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