Revision as of 15:45, 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 solors. 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 cardf 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 if $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .

Solution

$\boxed{013}$

there have to be 2 of 8 card sharing same number and 2 of them sharing same color.

and the there 2 pairs of cards can't be all same or there will be 2 card which are completely same

WLOG the number are 1,1,2,3,4,5,6,and7 and the color are a,a,b,c,d,e,f,andg then we can get 2 cases

1: 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

2: 1b,1c,2a,3a,4d,5e,6f,and 7g in this case, we can't discard. there are (6*5)/2=15 situations in this case

so the proprobility is 12/(12+15)=4/9

the answer is 4+9=013

2017 AIME II (Problems • Answer Key • Resources)
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Difference between revisions of "2017 AIME II Problems/Problem 9"

Revision as of 15:45, 23 March 2017

Problem

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@@ Line 4: / Line 4: @@
 ==Solution==
 <math>\boxed{013}</math>
 there have to be 2 of 8 card sharing same number and 2 of them sharing same color.
 and the there 2 pairs of cards can't be all same or there will be 2 card which are completely same
 WLOG the number are 1,1,2,3,4,5,6,and7 and the color are a,a,b,c,d,e,f,andg
 then we can get 2 cases