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Difference between revisions of "2007 AIME II Problems/Problem 10"

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(→‎Solution 2)
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these can be denoted as <math>+A+B</math>, <math>+A-B</math>,<math>-A+B</math>, and <math>-A-B</math>
 
these can be denoted as <math>+A+B</math>, <math>+A-B</math>,<math>-A+B</math>, and <math>-A-B</math>
  
−
we note that if all of the elements are in <math>+A+B</math>, <math>+A-B</math> and <math>-A-B</math>
+
we note that if all of the elements are in <math>+A+B</math>, <math>+A-B</math> or <math>-A-B</math>
 
we have that <math>B</math> is a subset of <math>A</math>
 
we have that <math>B</math> is a subset of <math>A</math>
 
which can happen in <math>\dfrac{3^6}{4^6}</math> ways
 
which can happen in <math>\dfrac{3^6}{4^6}</math> ways
  
−
similarly if the elements are in <math>+A-B</math>,<math>-A+B</math>, and <math>-A-B</math>
+
similarly if the elements are in <math>+A-B</math>,<math>-A+B</math>, or <math>-A-B</math>
 
we have  that <math>B</math> is a subset of <math>S-A</math>
 
we have  that <math>B</math> is a subset of <math>S-A</math>
 
which can happen in <math>\dfrac{3^6}{4^6}</math> ways as well
 
which can happen in <math>\dfrac{3^6}{4^6}</math> ways as well
  
−
but we need to make sure we don't over-count ways that are in both sets these are when <math>+A-B</math> and <math>-A-B</math>
+
but we need to make sure we don't over-count ways that are in both sets these are when <math>+A-B</math> or <math>-A-B</math>
 
which can happen in <math>\dfrac{2^6}{4^6}</math> ways
 
which can happen in <math>\dfrac{2^6}{4^6}</math> ways
 
so our probability is <math>\dfrac{2*3^6-2^6}{4^6}= \dfrac{3^6-2^5}{2^11}=\dfrac{697}{2^{11}}</math>.
 
so our probability is <math>\dfrac{2*3^6-2^6}{4^6}= \dfrac{3^6-2^5}{2^11}=\dfrac{697}{2^{11}}</math>.

Revision as of 18:05, 8 March 2014

Problem

Let $S$ be a set with six elements. Let $P$ be the set of all subsets of $S.$ Subsets $A$ and $B$ of $S$, not necessarily distinct, are chosen independently and at random from $P$. The probability that $B$ is contained in at least one of $A$ or $S-A$ is $\frac{m}{n^{r}},$ where $m$, $n$, and $r$ are positive integers, $n$ is prime, and $m$ and $n$ are relatively prime. Find $m+n+r.$ (The set $S-A$ is the set of all elements of $S$ which are not in $A.$)

Solution 1

Use casework:

  • $B$ has 6 elements:
    • Probability: $\frac{1}{2^6} = \frac{1}{64}$
    • $A$ must have either 0 or 6 elements, probability: $\frac{2}{2^6} = \frac{2}{64}$.
  • $B$ has 5 elements:
    • Probability: ${6\choose5}/64 = \frac{6}{64}$
    • $A$ must have either 0, 6, or 1, 5 elements. The total probability is $\frac{2}{64} + \frac{2}{64} = \frac{4}{64}$.
  • $B$ has 4 elements:
    • Probability: ${6\choose4}/64 = \frac{15}{64}$
    • $A$ must have either 0, 6; 1, 5; or 2,4 elements. If there are 1 or 5 elements, the set which contains 5 elements must have four emcompassing $B$ and a fifth element out of the remaining $2$ numbers. The total probability is $\frac{2}{64}\left({2\choose0} + {2\choose1} + {2\choose2}\right) = \frac{2}{64} + \frac{4}{64} + \frac{2}{64} = \frac{8}{64}$.

We could just continue our casework. In general, the probability of picking B with $n$ elements is $\frac{{6\choose n}}{64}$. Since the sum of the elements in the $k$th row of Pascal's Triangle is $2^k$, the probability of obtaining $A$ or $S-A$ which encompasses $B$ is $\frac{2^{7-n}}{64}$. In addition, we must count for when $B$ is the empty set (probability: $\frac{1}{64}$), of which all sets of $A$ will work (probability: $1$).

Thus, the solution we are looking for is $\left(\sum_{i=1}^6 \frac{{6\choose i}}{64} \cdot \frac{2^{7-i}}{64}\right) + \frac{1}{64} \cdot \frac{64}{64}<cmath>=\frac{(1)(64)+(6)(64)+(15)(32)+(20)(16)+(15)(8)+(6)(4)+(1)(2)}{(64)(64)}</cmath>=\frac{1394}{2^{12}}$$=\frac{697}{2^{11}}$.

The answer is $697 + 2 + 11 = 710$.

Solution 2

we need $B$ to be a subset of $A$ or $S-A$ we can divide each element of $S$ into 4 categories:

  • it is in $A$ and $B$
  • it is in $A$ but not in $B$
  • it is not in $A$ but is in $B$
  • or it is not in $A$ and not in $B$

these can be denoted as $+A+B$, $+A-B$,$-A+B$, and $-A-B$

we note that if all of the elements are in $+A+B$, $+A-B$ or $-A-B$ we have that $B$ is a subset of $A$ which can happen in $\dfrac{3^6}{4^6}$ ways

similarly if the elements are in $+A-B$,$-A+B$, or $-A-B$ we have that $B$ is a subset of $S-A$ which can happen in $\dfrac{3^6}{4^6}$ ways as well

but we need to make sure we don't over-count ways that are in both sets these are when $+A-B$ or $-A-B$ which can happen in $\dfrac{2^6}{4^6}$ ways so our probability is $\dfrac{2*3^6-2^6}{4^6}= \dfrac{3^6-2^5}{2^11}=\dfrac{697}{2^{11}}$.

so the final answer is $697 + 2 + 11 = 710$.

See also

2007 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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