2005 AMC 12B Problems/Problem 11
- The following problem is from both the 2005 AMC 12B #11 and 2005 AMC 10B #15, so both problems redirect to this page.
Contents
Problem
An envelope contains eight bills: ones,
fives,
tens, and
twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $
or more?
Solution 1
The only way to get a total of $ or more is if you pick a twenty and another bill, or if you pick both tens. There are a total of
ways to choose
bills out of
. There are
ways to choose a twenty and some other non-twenty bill. There is
way to choose both twenties, and also
way to choose both tens. Adding these up, we find that there are a total of
ways to attain a sum of
or greater, so there is a total probability of
.
Solution 2
Another way to do this problem is to use complementary counting, i.e. how many ways that the sum is less than . Now, you do not have to consider the
twenties, so you have
bills left.
ways. However, you counted the case when you have
tens, so you need to subtract 1, and you get
. Finding the ways to get
or higher, you subtract
from
and get
. So the answer is
Solution 3
There are two cases that work, namely getting at least twenty, or getting
tens.
Case :
Case :
Summing up our cases, we have
Solution 4
Note that if a twenty is drawn, anything else that is drawn will create a total greater than ; The probability of a twenty being drawn first is
The same could be said for drawing anything, and then drawing a twenty. However, we can only draw something that isn't a twenty first (since we've already accounted for the probability of drawing two twenties).
The probability of drawing a non-twenty first, then a twenty second is Finally, we can draw two tens. The probability of this occuring is
Adding these three probabilities gives us
Video Solution by WhyMath
~savannahsolver
See also
2005 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2005 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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