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Difference between revisions of "2017 AIME II Problems/Problem 9"
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A special deck of cards contains <math>49</math> cards, each labeled with a number from <math>1</math> to <math>7</math> and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and <math>\textit{still}</math> have at least one card of each color and at least one card with each number is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
 
A special deck of cards contains <math>49</math> cards, each labeled with a number from <math>1</math> to <math>7</math> and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and <math>\textit{still}</math> have at least one card of each color and at least one card with each number is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
  
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==Solution==
+
==Solution 1==
 +
Without loss of generality, assume that the <math>8</math> numbers on Sharon's cards are <math>1</math>, <math>1</math>, <math>2</math>, <math>3</math>, <math>4</math>, <math>5</math>, <math>6</math>, and <math>7</math>, in that order, and assume the <math>8</math> colors are red, red, and five different arbituary colors. There are <math>{8\choose2}-1</math> ways of assigning the two red cards to the <math>8</math> numbers; we subtract <math>1</math> because we cannot assign the two reds to the two <math>1</math>'s.
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In order for Sharon to be able to remove at least one card and still have at least one card of each color, one of the reds have to be assigned with one of the <math>1</math>s. The number of ways for this not to happen is <math>{6\choose2}</math>, so the number of ways for it to happen is <math>\left({8\choose2}-1\right)-{6\choose2}</math>. Each of these assignments is equally likely, so the probability that Sharn can discard one of her cards and still have at least one card of each color and at least one card with each number is <math>\frac{\left({8\choose2}-1\right)-{6\choose2}}{{8\choose2}-1}=\frac{4}{9}</math>, and <math>4+9=\boxed{13}</math>.
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==Solution 2==
  
 
There have to be <math>2</math> of <math>8</math> cards sharing the same number and <math>2</math> of them sharing same color.
 
There have to be <math>2</math> of <math>8</math> cards sharing the same number and <math>2</math> of them sharing same color.

Revision as of 16:51, 23 March 2017

Problem

A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and $\textit{still}$ have at least one card of each color and at least one card with each number is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Solution 1

Without loss of generality, assume that the $8$ numbers on Sharon's cards are $1$, $1$, $2$, $3$, $4$, $5$, $6$, and $7$, in that order, and assume the $8$ colors are red, red, and five different arbituary colors. There are ${8\choose2}-1$ ways of assigning the two red cards to the $8$ numbers; we subtract $1$ because we cannot assign the two reds to the two $1$'s. In order for Sharon to be able to remove at least one card and still have at least one card of each color, one of the reds have to be assigned with one of the $1$s. The number of ways for this not to happen is ${6\choose2}$, so the number of ways for it to happen is $\left({8\choose2}-1\right)-{6\choose2}$. Each of these assignments is equally likely, so the probability that Sharn can discard one of her cards and still have at least one card of each color and at least one card with each number is $\frac{\left({8\choose2}-1\right)-{6\choose2}}{{8\choose2}-1}=\frac{4}{9}$, and $4+9=\boxed{13}$.

Solution 2

There have to be $2$ of $8$ cards sharing the same number and $2$ of them sharing same color.

$2$ pairs of cards can't be the same or else there will be $2$ card which are completely same.

WLOG the numbers are $1,1,2,3,4,5,6,$ and $7$ and the colors are $a,a,b,c,d,e,f,$ and $g$ Then we can get $2$ cases:

Case One: $1a,1b,2a,3c,4d,5e,6f,$ and $7g$ in this case, we can discard $1a$. there are $2*6=12$ situations in this case.

Case Two: $1b,1c,2a,3a,4d,5e,6f,$ and $7g$ In this case, we can't discard. There are $\frac{6*5}{2}=15$ situations in this case

So the probability is $\frac{12}{12+15}=\frac{4}{9}$

The answer is $4+9=\boxed{013}$

See Also

2017 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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