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Difference between revisions of "2011 AMC 10A Problems/Problem 21"

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== Problem 21 ==
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== Problem ==
 
Two counterfeit coins of equal weight are mixed with <math>8</math> identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the <math>10</math> coins. A second pair is selected at random without replacement from the remaining <math>8</math> coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all <math>4</math> selected coins are genuine?
 
Two counterfeit coins of equal weight are mixed with <math>8</math> identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the <math>10</math> coins. A second pair is selected at random without replacement from the remaining <math>8</math> coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all <math>4</math> selected coins are genuine?
  
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== Solution 1==
 
== Solution 1==
  
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Note that we are trying to find the conditional probability <math>P(A \vert B) = \frac{P(A \cap B)}{P(B)}</math> where <math>A</math> is the <math>4</math> coins being genuine and <math>B</math> is the sum of the weight of the coins being equal. The only possibilities for <math>B</math> are (g is abbreviated for genuine and c is abbreviated for counterfeit, and the pairs are the first and last two in the quadruple) <math>(g,g,g,g),(g,c,g,c),(g,c,c,g),(c,g,g,c),(c,g,c,g)</math>. We see that <math>A \cap B</math> happens with probability <math>\frac{8}{10} \times \frac{7}{9} \times \frac{6}{8} \times \frac{5}{7} = \frac{1}{3}</math>, and <math>B</math> happens with probability <math>\frac{1}{3}+4 \times \left( \frac{8}{10} \times \frac{2}{9} \times \frac{7}{8}\times \frac{1}{7}\right) = \frac{19}{45}</math>, hence <math>P(A \vert B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{3}}{\frac{19}{45}}=\boxed{\frac{15}{19} \ \mathbf{(D)}}</math>.
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If we pick <math>4</math> indistinguishable real coins from the set of <math>8</math> real coins, there are <math>\binom{8}{4}</math> ways to pick the coins.  If we then place the coins in four distinguishable slots on the scale, there are <math>4!</math> ways to arrange them, giving <math>4!\cdot \binom{8}{4}</math> ways to choose and place <math>8</math> real coins. This gives <math>1680</math> desirable combinations.
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If we pick <math>2</math> real coins and <math>2</math> fake coins, there are <math>\binom{8}{2}\binom{2}{2}</math> ways to choose the coins.  There are <math>4</math> choices for the first slot on the left side of the scale.  Whichever type of coin is placed in that first slot, there are <math>2</math> choices for the second slot on the left side of the scale, since it must be of the opposite type of coin.  There are <math>2</math> choices for the first slot on the right side of the scale and only <math>1</math> choice for the last slot on the right side.
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Thus, there are <math>4\cdot 2\cdot 2\cdot 1 = 16</math> ways to arrange the coins, and <math>\binom{8}{2}\binom{2}{2} = 28</math> sets of possible coins, for a total of <math>16\cdot 28 = 448</math> combinations that are legal, yet undesirable.
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The overall probability is thus <math>\frac{1680}{1680 + 448} = \boxed{\frac{15}{19} \ \mathbf{(D)}}</math>.
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Note that in this solution, all four slots on the scale are deemed to be distinguishable.  In essence, this "overcounts" all numbers by a factor of <math>8</math>, since you can switch the coins on the left side, you can switch the coins on the right side, or you can switch sides of the scale.  But since all numbers are increased 8-fold, it will cancel out when calculating the probability.
  
 
== Solution 2==
 
== Solution 2==
  
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If we pick <math>4</math> distinguishable real coins from the set of <math>8</math> real coins, there are <math>\binom{8}{4}</math> ways to pick the coins. If we then place the coins in four distinguishable slots on the scale, there are <math>4!</math> ways to arrange them, giving <math>4!\cdot \binom{8}{4}</math> ways to choose and place <math>8</math> real coins.  This gives <math>1680</math> desirable combinations.
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Place the two coins from the first pair in a line followed by the two coins from the second pair followed by the six left-over coins.  We can do that in <math>\binom{10}{2} = 45</math> different ways.
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We need to exclude those cases where the weight shows a difference.  There are two cases where a pair has both counterfeit coins and there are <math>4\cdot6</math> cases where one counterfeit coin is chosen and one is in the left-over.  That leaves <math>45-2-24=19</math> cases.
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Of these, <math>\binom{6}{2}=15</math> cases has both counterfeit coins in the left-over.  The probability of having chosen four genuine coins therefore is <math>\boxed{\frac{15}{19} \ \mathbf{(D)}}</math>.
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== Solution 3==
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WLOG, allow for all the coins to be distinguishable.
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We split this up into cases. Case <math>1</math> being the weight of <math>2</math> genuine coins together and Case <math>2</math> being the weight of <math>1</math> genuine coin and <math>1</math> counterfeit coin.
  
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If we pick <math>2</math> real coins and <math>2</math> fake coins, there are <math>\binom{8}{2}\binom{2}{2}</math> ways to choose the coins.  There are <math>4</math> choices for the first slot on the left side of the scale.  Whichever type of coin is placed in that first slot, there are <math>2</math> choices for the second slot on the left side of the scale, since it must be of the opposite type of coin.  There are <math>2</math> choices for the first slot on the right side of the scale, and only <math>1</math> choice for the last slot on the right side.
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Case <math>1</math>: All Genuine coins chosen.
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This happens in <math>\frac{\binom{8}{2}\cdot\binom{6}{2}}{2}=210</math> ways
  
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Thus, there are <math>4\cdot 2\cdot 2\cdot 1 = 16</math> ways to arrange the coins, and <math>\binom{8}{2}\binom{2}{2} = 28</math> sets of possible coins, for a total of <math>16\cdot 28 = 448</math> combinations that are legal, yet undesirable.
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Case <math>2</math>: Genuine coin and Counterfeit coin both chosen.
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This happens in <math>8\cdot7=56</math> ways.
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Hence, the answer is <math>\boxed{\frac{15}{19} \ \mathbf{(D)}}</math>.
  
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The overall probability is thus <math>\frac{1680}{1680 + 448} = \boxed{\frac{15}{19} \ \mathbf{(D)}}</math>.
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==Video Solution by the Beauty of Math==
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https://www.youtube.com/watch?v=W5NlV2B_83U&feature=push-u-sub&attr_tag=XAAsHxmGZAHEdZgT%3A6
  
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Note that in this solution, all four slots on the scale are deemed to be distinguishable.  In essence, this "overcounts" all numbers by a factor of <math>8</math>, since you can switch the coins on the left side, you can switch the coins on the right side, or you can switch sides of the scale.  But since all numbers are increased 8-fold, it will cancel out when calculating the probability.
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~Icematrix
  
 
== See Also ==
 
== See Also ==
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[[Category:Introductory Combinatorics Problems]]
 
[[Category:Introductory Combinatorics Problems]]
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{{MAA Notice}}

Latest revision as of 14:04, 9 June 2024

Problem

Two counterfeit coins of equal weight are mixed with $8$ identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the $10$ coins. A second pair is selected at random without replacement from the remaining $8$ coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all $4$ selected coins are genuine?

$\textbf{(A)}\ \frac{7}{11}\qquad\textbf{(B)}\ \frac{9}{13}\qquad\textbf{(C)}\ \frac{11}{15}\qquad\textbf{(D)}\ \frac{15}{19}\qquad\textbf{(E)}\ \frac{15}{16}$

Solution 1

If we pick $4$ indistinguishable real coins from the set of $8$ real coins, there are $\binom{8}{4}$ ways to pick the coins. If we then place the coins in four distinguishable slots on the scale, there are $4!$ ways to arrange them, giving $4!\cdot \binom{8}{4}$ ways to choose and place $8$ real coins. This gives $1680$ desirable combinations.

If we pick $2$ real coins and $2$ fake coins, there are $\binom{8}{2}\binom{2}{2}$ ways to choose the coins. There are $4$ choices for the first slot on the left side of the scale. Whichever type of coin is placed in that first slot, there are $2$ choices for the second slot on the left side of the scale, since it must be of the opposite type of coin. There are $2$ choices for the first slot on the right side of the scale and only $1$ choice for the last slot on the right side.

Thus, there are $4\cdot 2\cdot 2\cdot 1 = 16$ ways to arrange the coins, and $\binom{8}{2}\binom{2}{2} = 28$ sets of possible coins, for a total of $16\cdot 28 = 448$ combinations that are legal, yet undesirable.

The overall probability is thus $\frac{1680}{1680 + 448} = \boxed{\frac{15}{19} \ \mathbf{(D)}}$.

Note that in this solution, all four slots on the scale are deemed to be distinguishable. In essence, this "overcounts" all numbers by a factor of $8$, since you can switch the coins on the left side, you can switch the coins on the right side, or you can switch sides of the scale. But since all numbers are increased 8-fold, it will cancel out when calculating the probability.

Solution 2

Place the two coins from the first pair in a line followed by the two coins from the second pair followed by the six left-over coins. We can do that in $\binom{10}{2} = 45$ different ways.

We need to exclude those cases where the weight shows a difference. There are two cases where a pair has both counterfeit coins and there are $4\cdot6$ cases where one counterfeit coin is chosen and one is in the left-over. That leaves $45-2-24=19$ cases.

Of these, $\binom{6}{2}=15$ cases has both counterfeit coins in the left-over. The probability of having chosen four genuine coins therefore is $\boxed{\frac{15}{19} \ \mathbf{(D)}}$.

Solution 3

WLOG, allow for all the coins to be distinguishable. We split this up into cases. Case $1$ being the weight of $2$ genuine coins together and Case $2$ being the weight of $1$ genuine coin and $1$ counterfeit coin.

Case $1$: All Genuine coins chosen. This happens in $\frac{\binom{8}{2}\cdot\binom{6}{2}}{2}=210$ ways

Case $2$: Genuine coin and Counterfeit coin both chosen. This happens in $8\cdot7=56$ ways.

Hence, the answer is $\boxed{\frac{15}{19} \ \mathbf{(D)}}$.

Video Solution by the Beauty of Math

https://www.youtube.com/watch?v=W5NlV2B_83U&feature=push-u-sub&attr_tag=XAAsHxmGZAHEdZgT%3A6

~Icematrix

See Also

2011 AMC 10A (Problems • Answer Key • Resources)
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