Difference between revisions of "2007 AIME II Problems/Problem 10"
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== Problem == | == Problem == | ||
− | Let <math>S</math> be a set with six | + | Let <math>S</math> be a [[set]] with six [[element]]s. Let <math>P</math> be the set of all [[subset]]s of <math>S.</math> Subsets <math>A</math> and <math>B</math> of <math>S</math>, not necessarily distinct, are chosen independently and at random from <math>P</math>. The [[probability]] that <math>B</math> is contained in at least one of <math>A</math> or <math>S-A</math> is <math>\frac{m}{n^{r}},</math> where <math>m</math>, <math>n</math>, and <math>r</math> are [[positive]] [[integer]]s, <math>n</math> is [[prime]], and <math>m</math> and <math>n</math> are [[relatively prime]]. Find <math>m+n+r.</math> (The set <math>S-A</math> is the set of all elements of <math>S</math> which are not in <math>A.</math>) |
== Solution == | == Solution == |
Revision as of 16:42, 30 March 2007
Problem
Let be a set with six elements. Let
be the set of all subsets of
Subsets
and
of
, not necessarily distinct, are chosen independently and at random from
. The probability that
is contained in at least one of
or
is
where
,
, and
are positive integers,
is prime, and
and
are relatively prime. Find
(The set
is the set of all elements of
which are not in
)
Solution
Use casework:
has 6 elements:
- Probability:
must have either 0 or 6 elements, probability:
.
- Probability:
has 5 elements:
- Probability:
must have either 0, 6, or 1, 5 elements. The total probability is
.
- Probability:
has 4 elements:
- Probability:
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
and a fifth element out of the remaining
numbers. The total probability is
.
- Probability:
We could just continue our casework. In general, the probability of picking B with elements is
. Since the sum of the elements in the
th row of Pascal's Triangle is
, the probability of obtaining
or
which encompasses
is
. In addition, we must count for when
is the empty set (probability:
), of which all sets of
will work (probability:
).
Thus, the solution we are looking for is .
The answer is .
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 |