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

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

@@ Line 19: / Line 19: @@
 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>:
-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>6^{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
+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
-<cmath>\sum_{k=1}^6 k^{6-k} = 6\cdot 1 + 15\cdot 16 + 20\cdot 27 + 15\cdot 16 + 6\cdot 5 + 1) = 1057</cmath>
+<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>
 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>

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