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Difference between revisions of "2017 AIME I Problems/Problem 7"

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-rocketscience
 
-rocketscience
  
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==Solution 1 but different==
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==Solution 1 but different (Committee Forming)==
 
Alternatively, one can note that we can consider groups where <math>a+b</math> is constant, say <math>c</math>. Fix any value of <math>c</math>. Then the sum of all of the values of <math>T(a,b)</math> such that <math>a+b=c</math> is <math>\binom{6}{a+b} \sum_{a+b=c} \binom{6}{a}\binom{6}{b}</math> which by Vandermonde's is <math>\binom{6}{a+b}\binom{12}{a+b}</math>. Remember, that expression is the resulting sum for a fixed <math>a+b</math>. So, for <math>a+b\le 6</math>, we want <math>\sum_{c=0}^{6} \binom{6}{c}\binom{12}{c}</math>. This is (by Vandermonde's or committee forming) <math>\binom{18}{6} = 18564 \implies \boxed{564}</math> ~ firebolt360
 
Alternatively, one can note that we can consider groups where <math>a+b</math> is constant, say <math>c</math>. Fix any value of <math>c</math>. Then the sum of all of the values of <math>T(a,b)</math> such that <math>a+b=c</math> is <math>\binom{6}{a+b} \sum_{a+b=c} \binom{6}{a}\binom{6}{b}</math> which by Vandermonde's is <math>\binom{6}{a+b}\binom{12}{a+b}</math>. Remember, that expression is the resulting sum for a fixed <math>a+b</math>. So, for <math>a+b\le 6</math>, we want <math>\sum_{c=0}^{6} \binom{6}{c}\binom{12}{c}</math>. This is (by Vandermonde's or committee forming) <math>\binom{18}{6} = 18564 \implies \boxed{564}</math> ~ firebolt360
 
===Note===
 
===Note===
 
Now just a quick explanation for people who don't fully understand Vandermonde's. Take the first part, <math>\sum_{a+b=c} \binom{6}{a}\binom{6}{b}</math>. Consider <math>2</math> different groups, <math>A</math> and <math>B</math> both of size <math>6</math> people. We wish to chose <math>a</math> peoples from <math>A</math> and <math>b=c-a</math> people from <math>B</math>. In total, we chose <math>c-a+a=c</math> people. We can then draw a bijection towards choosing <math>c</math> people from <math>A\cup B</math>, which has size <math>12</math>. So, it is <math>\binom{12}{c}=\binom{12}{a+b}</math>. Similarly, for <math>\sum_{c=6} \binom{6}{c}\binom{12}{c}</math>, we see that <math>\binom{6}{c}=\binom{6}{6-c}</math>. Now the total is <math>18</math>, and the sum is <math>6</math>. So, we get <math>\binom{18}{6}</math>. See [[committee forming]] for more information ~ firebolt360
 
Now just a quick explanation for people who don't fully understand Vandermonde's. Take the first part, <math>\sum_{a+b=c} \binom{6}{a}\binom{6}{b}</math>. Consider <math>2</math> different groups, <math>A</math> and <math>B</math> both of size <math>6</math> people. We wish to chose <math>a</math> peoples from <math>A</math> and <math>b=c-a</math> people from <math>B</math>. In total, we chose <math>c-a+a=c</math> people. We can then draw a bijection towards choosing <math>c</math> people from <math>A\cup B</math>, which has size <math>12</math>. So, it is <math>\binom{12}{c}=\binom{12}{a+b}</math>. Similarly, for <math>\sum_{c=6} \binom{6}{c}\binom{12}{c}</math>, we see that <math>\binom{6}{c}=\binom{6}{6-c}</math>. Now the total is <math>18</math>, and the sum is <math>6</math>. So, we get <math>\binom{18}{6}</math>. See [[committee forming]] for more information ~ firebolt360
  
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== Solution 2 ==  
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== Solution 2 (Committee Forming but slightly more bashy)==  
  
 
Treating <math>a+b</math> as <math>n</math>, this problem asks for <cmath>\sum_{n=0}^{6} \left[\binom{6}{n}\sum_{m=0}^{n}\left[\binom{6}{m}\binom{6}{n-m}\right]\right].</cmath> But <cmath>\sum_{m=0}^{n} \left[\binom{6}{m} \binom{6}{n-m}\right]</cmath>
 
Treating <math>a+b</math> as <math>n</math>, this problem asks for <cmath>\sum_{n=0}^{6} \left[\binom{6}{n}\sum_{m=0}^{n}\left[\binom{6}{m}\binom{6}{n-m}\right]\right].</cmath> But <cmath>\sum_{m=0}^{n} \left[\binom{6}{m} \binom{6}{n-m}\right]</cmath>
 
can be computed through the following combinatorial argument.  Choosing <math>n</math> elements from a set of size <math>12</math> is the same as splitting the set into two sets of size <math>6</math> and choosing <math>m</math> elements from one, <math>n-m</math> from the other where <math>0\leq m\leq n</math>.  The number of ways to perform such a procedure is simply <math>\binom{12}{n}</math>. Therefore, the requested sum is <cmath>\sum_{n=0}^{6} \left[\binom{6}{n} \binom{12}{n}\right] = 18564.</cmath> As such, our answer is <math>\boxed{564}</math>.
 
can be computed through the following combinatorial argument.  Choosing <math>n</math> elements from a set of size <math>12</math> is the same as splitting the set into two sets of size <math>6</math> and choosing <math>m</math> elements from one, <math>n-m</math> from the other where <math>0\leq m\leq n</math>.  The number of ways to perform such a procedure is simply <math>\binom{12}{n}</math>. Therefore, the requested sum is <cmath>\sum_{n=0}^{6} \left[\binom{6}{n} \binom{12}{n}\right] = 18564.</cmath> As such, our answer is <math>\boxed{564}</math>.
  
 +
Note from epiconan:
 +
To avoid the bash, you can actually use Vandermondes <math>\emph{again}</math>.
 +
<cmath>\sum_{n=0}^{6} \left[\binom{6}{n} \binom{12}{n}\right] = \sum_{n=0}^{6} \left[\binom{6}{6-n} \binom{12}{n}\right] = \binom{6+12}{6} = \binom{18}{6}.</cmath>
 
- Awsomness2000
 
- Awsomness2000
  
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- mathymath
 
- mathymath
  
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==Solution 5==
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==Solution 5 (Committee Forming but different)==
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Let <math>c=6-(a+b)</math>. Then <math>\binom{6}{a+b}=\binom{6}{c}</math>, and <math>a+b+c=6</math>. The problem thus asks for <cmath>\sum_{a+b+c=6}\binom{6}{a}\binom{6}{b}\binom{6}{c} \pmod {1000}.</cmath> Suppose we have <math>6</math> red balls, <math>6</math> green balls, and <math>6</math> blue balls lined up in a row, and we want to choose <math>6</math> balls from this set of <math>18</math> balls by considering each color separately. Over all possible selections of <math>6</math> balls from this set, there are always a nonnegative number of balls in each color group. The answer is <math>\binom{18}{6} \pmod {1000}=18\boxed{564}</math>.  
+
Let <math>c=6-(a+b)</math>. Then <math>\binom{6}{a+b}=\binom{6}{c}</math>, and <math>a+b+c=6</math>. The problem thus asks for <cmath>\sum_{a+b+c=6}\binom{6}{a}\binom{6}{b}\binom{6}{c} \pmod {1000}.</cmath> Suppose we have <math>6</math> red balls, <math>6</math> green balls, and <math>6</math> blue balls lined up in a row, and we want to choose <math>6</math> balls from this set of <math>18</math> balls by considering each color separately. Over all possible selections of <math>6</math> balls from this set, there are always a nonnegative number of balls in each color group. The answer is <math>\binom{18}{6} \pmod {1000}=18\boxed{564}</math>.
  
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==Solution 6==
+
==Solution 5 but different (Committee Forming)==
 
Since <math>\binom{6}{n}=\binom{6}{6-n}</math>, we can rewrite <math>T(a,b)</math> as <math>\binom{6}{a}\binom{6}{b}\binom{6}{6-(a+b)}</math>. Consider the number of ways to choose a committee of 6 people from a group of 6 democrats, 6 republicans, and 6 independents. We can first pick <math>a</math> democrats, then pick <math>b</math> republicans, provided that <math>a+b \leq 6</math>. Then we can pick the remaining <math>6-(a+b)</math> people from the independents. But this is just <math>T(a,b)</math>, so the sum of all <math>T(a,b)</math> is equal to the number of ways to choose this committee.
 
Since <math>\binom{6}{n}=\binom{6}{6-n}</math>, we can rewrite <math>T(a,b)</math> as <math>\binom{6}{a}\binom{6}{b}\binom{6}{6-(a+b)}</math>. Consider the number of ways to choose a committee of 6 people from a group of 6 democrats, 6 republicans, and 6 independents. We can first pick <math>a</math> democrats, then pick <math>b</math> republicans, provided that <math>a+b \leq 6</math>. Then we can pick the remaining <math>6-(a+b)</math> people from the independents. But this is just <math>T(a,b)</math>, so the sum of all <math>T(a,b)</math> is equal to the number of ways to choose this committee.
 
On the other hand, we can simply pick any 6 people from the <math>6+6+6=18</math> total politicians in the group. Clearly, there are <math>\binom{18}{6}</math> ways to do this. So the desired quantity is equal to <math>\binom{18}{6}</math>. We can then compute (routinely) the last 3 digits of <math>\binom{18}{6}</math> as <math>\boxed{564}</math>.
 
On the other hand, we can simply pick any 6 people from the <math>6+6+6=18</math> total politicians in the group. Clearly, there are <math>\binom{18}{6}</math> ways to do this. So the desired quantity is equal to <math>\binom{18}{6}</math>. We can then compute (routinely) the last 3 digits of <math>\binom{18}{6}</math> as <math>\boxed{564}</math>.
 +
 +
==Solution 6(NICE Journal) ==
 +
Note that <math>\binom{6}{a+b} = \binom{6}{6-a-b}</math>. So we have <math>\binom{6}{a}\binom{6}{b}\binom{6}{6-a-b}</math>. If we think about this this is essentially choosing a group of <math>a</math> people from <math>6</math> people, a group of <math>b</math> people from <math>6</math> people, and a group of <math>6 - a -b</math> from another group of <math>6</math> people. This is nothing but choosing <math>6</math> people from a group of <math>18</math> people. This is nothing but <math>\binom{18}{6} = 18564 \Rightarrow 564</math>.
 +
~coolmath_2018
 +
 +
==Remark ==
 +
This problem is an example of the generalization of Vandermonde's theorem, which states that for nonnegative <math>k_1, k_2, \ldots k_p</math> and <math>n_1, n_2, \ldots n_p</math>, we have
 +
<cmath>\sum _{k_1+\cdots +k_p=m}\binom{n_1}{k_1} \binom{n_2}{k_2} \cdots \binom{n_p}{k_p} = \binom{n_1+ \cdots +n_p}{m}.</cmath>
 +
~eibc
  
 
==See Also==
 
==See Also==
 
{{AIME box|year=2017|n=I|num-b=6|num-a=8}}
 
{{AIME box|year=2017|n=I|num-b=6|num-a=8}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 22:42, 8 December 2024

Problem 7

For nonnegative integers $a$ and $b$ with $a + b \leq 6$, let $T(a, b) = \binom{6}{a} \binom{6}{b} \binom{6}{a + b}$. Let $S$ denote the sum of all $T(a, b)$, where $a$ and $b$ are nonnegative integers with $a + b \leq 6$. Find the remainder when $S$ is divided by $1000$.

Major Note

Most solutions use committee forming (except for the bash solution). To understand more about the techniques used, visit the committee forming page for more information.

Solution 1 (Committee Forming)

Let $c=6-(a+b)$, and note that $\binom{6}{a + b}=\binom{6}{c}$. The problem thus asks for the sum $\binom{6}{a} \binom{6}{b} \binom{6}{c}$ over all $a,b,c$ such that $a+b+c=6$. Consider an array of 18 dots, with 3 columns of 6 dots each. The desired expression counts the total number of ways to select 6 dots by considering each column separately, which is equal to $\binom{18}{6}=18564$. Therefore, the answer is $\boxed{564}$.

-rocketscience

Solution 1 but different (Committee Forming)

Alternatively, one can note that we can consider groups where $a+b$ is constant, say $c$. Fix any value of $c$. Then the sum of all of the values of $T(a,b)$ such that $a+b=c$ is $\binom{6}{a+b} \sum_{a+b=c} \binom{6}{a}\binom{6}{b}$ which by Vandermonde's is $\binom{6}{a+b}\binom{12}{a+b}$. Remember, that expression is the resulting sum for a fixed $a+b$. So, for $a+b\le 6$, we want $\sum_{c=0}^{6} \binom{6}{c}\binom{12}{c}$. This is (by Vandermonde's or committee forming) $\binom{18}{6} = 18564 \implies \boxed{564}$ ~ firebolt360

Note

Now just a quick explanation for people who don't fully understand Vandermonde's. Take the first part, $\sum_{a+b=c} \binom{6}{a}\binom{6}{b}$. Consider $2$ different groups, $A$ and $B$ both of size $6$ people. We wish to chose $a$ peoples from $A$ and $b=c-a$ people from $B$. In total, we chose $c-a+a=c$ people. We can then draw a bijection towards choosing $c$ people from $A\cup B$, which has size $12$. So, it is $\binom{12}{c}=\binom{12}{a+b}$. Similarly, for $\sum_{c=6} \binom{6}{c}\binom{12}{c}$, we see that $\binom{6}{c}=\binom{6}{6-c}$. Now the total is $18$, and the sum is $6$. So, we get $\binom{18}{6}$. See committee forming for more information ~ firebolt360

Solution 2 (Committee Forming but slightly more bashy)

Treating $a+b$ as $n$, this problem asks for \[\sum_{n=0}^{6} \left[\binom{6}{n}\sum_{m=0}^{n}\left[\binom{6}{m}\binom{6}{n-m}\right]\right].\] But \[\sum_{m=0}^{n} \left[\binom{6}{m} \binom{6}{n-m}\right]\] can be computed through the following combinatorial argument. Choosing $n$ elements from a set of size $12$ is the same as splitting the set into two sets of size $6$ and choosing $m$ elements from one, $n-m$ from the other where $0\leq m\leq n$. The number of ways to perform such a procedure is simply $\binom{12}{n}$. Therefore, the requested sum is \[\sum_{n=0}^{6} \left[\binom{6}{n} \binom{12}{n}\right] = 18564.\] As such, our answer is $\boxed{564}$.

Note from epiconan: To avoid the bash, you can actually use Vandermondes $\emph{again}$. \[\sum_{n=0}^{6} \left[\binom{6}{n} \binom{12}{n}\right] = \sum_{n=0}^{6} \left[\binom{6}{6-n} \binom{12}{n}\right] = \binom{6+12}{6} = \binom{18}{6}.\] - Awsomness2000

Solution 3 (Major Major Bash)

Case 1: $a<b$.

Subcase 1: $a=0$ \[\binom{6}{0}\binom{6}{1}\binom{6}{1}=36\] \[\binom{6}{0}\binom{6}{2}\binom{6}{2}=225\] \[\binom{6}{0}\binom{6}{3}\binom{6}{3}=400\] \[\binom{6}{0}\binom{6}{4}\binom{6}{4}=225\] \[\binom{6}{0}\binom{6}{5}\binom{6}{5}=36\] \[\binom{6}{0}\binom{6}{6}\binom{6}{6}=1\] \[36+225+400+225+36+1=923\] Subcase 2: $a=1$ \[\binom{6}{1}\binom{6}{2}\binom{6}{3}=1800 \equiv 800 \pmod {1000}\] \[\binom{6}{1}\binom{6}{3}\binom{6}{4}=1800 \equiv 800 \pmod {1000}\] \[\binom{6}{1}\binom{6}{4}\binom{6}{5}=540\] \[\binom{6}{1}\binom{6}{5}\binom{6}{6}=36\] \[800+800+540+36=2176 \equiv 176 \pmod {1000}\] Subcase 3: $a=2$ \[\binom{6}{2}\binom{6}{3}\binom{6}{5}=1800\equiv800\pmod{1000}\] \[\binom{6}{2}\binom{6}{4}\binom{6}{6}=225\] \[800+225=1025\equiv25\pmod{1000}\]


\[923+176+25=1124\equiv124\pmod{1000}\]

Case 2: $b<a$

By just switching $a$ and $b$ in all of the above cases, we will get all of the cases such that $b>a$ is true. Therefore, this case is also $124\pmod{1000}$

Case 3: $a=b$ \[\binom{6}{0}\binom{6}{0}\binom{6}{0}=1\] \[\binom{6}{1}\binom{6}{1}\binom{6}{2}=540\] \[\binom{6}{2}\binom{6}{2}\binom{6}{4}=3375\equiv375\pmod{1000}\] \[\binom{6}{3}\binom{6}{3}\binom{6}{6}=400\] \[1+540+375+400=1316\equiv316\pmod{1000}\]


\[316+124+124=\boxed{\boxed{564}}\]

Solution 4

We begin as in solution 1 to rewrite the sum as $\binom{6}{a} \binom{6}{b} \binom{6}{c}$ over all $a,b,c$ such that $a+b+c=6$. Consider the polynomial $P(x)=\left(\binom{6}{0} + \binom{6}{1}x + \binom{6}{2}x^2 + \binom{6}{3}x^3 + \cdot \cdot \cdot + \binom{6}{6}x^6\right)^3$. We can see the sum we wish to compute is just the coefficient of the $x^6$ term. However $P(x)=((1+x)^6)^3=(1+x)^{18}$. Therefore, the coefficient of the $x^6$ term is just $\binom{18}{6} = 18564$ so the answer is $\boxed{564}$.

- mathymath

Solution 5 (Committee Forming but different)

Let $c=6-(a+b)$. Then $\binom{6}{a+b}=\binom{6}{c}$, and $a+b+c=6$. The problem thus asks for \[\sum_{a+b+c=6}\binom{6}{a}\binom{6}{b}\binom{6}{c} \pmod {1000}.\] Suppose we have $6$ red balls, $6$ green balls, and $6$ blue balls lined up in a row, and we want to choose $6$ balls from this set of $18$ balls by considering each color separately. Over all possible selections of $6$ balls from this set, there are always a nonnegative number of balls in each color group. The answer is $\binom{18}{6} \pmod {1000}=18\boxed{564}$.

Solution 5 but different (Committee Forming)

Since $\binom{6}{n}=\binom{6}{6-n}$, we can rewrite $T(a,b)$ as $\binom{6}{a}\binom{6}{b}\binom{6}{6-(a+b)}$. Consider the number of ways to choose a committee of 6 people from a group of 6 democrats, 6 republicans, and 6 independents. We can first pick $a$ democrats, then pick $b$ republicans, provided that $a+b \leq 6$. Then we can pick the remaining $6-(a+b)$ people from the independents. But this is just $T(a,b)$, so the sum of all $T(a,b)$ is equal to the number of ways to choose this committee. On the other hand, we can simply pick any 6 people from the $6+6+6=18$ total politicians in the group. Clearly, there are $\binom{18}{6}$ ways to do this. So the desired quantity is equal to $\binom{18}{6}$. We can then compute (routinely) the last 3 digits of $\binom{18}{6}$ as $\boxed{564}$.

Solution 6(NICE Journal)

Note that $\binom{6}{a+b} = \binom{6}{6-a-b}$. So we have $\binom{6}{a}\binom{6}{b}\binom{6}{6-a-b}$. If we think about this this is essentially choosing a group of $a$ people from $6$ people, a group of $b$ people from $6$ people, and a group of $6 - a -b$ from another group of $6$ people. This is nothing but choosing $6$ people from a group of $18$ people. This is nothing but $\binom{18}{6} = 18564 \Rightarrow 564$. ~coolmath_2018

Remark

This problem is an example of the generalization of Vandermonde's theorem, which states that for nonnegative $k_1, k_2, \ldots k_p$ and $n_1, n_2, \ldots n_p$, we have \[\sum _{k_1+\cdots +k_p=m}\binom{n_1}{k_1} \binom{n_2}{k_2} \cdots \binom{n_p}{k_p} = \binom{n_1+ \cdots +n_p}{m}.\] ~eibc

See Also

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