2017 AIME II Problems/Problem 9
Problem
A special deck of cards contains 
cards, each labeled with a number from 
to 
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 
have at least one card of each color and at least one card with each number if 
, where 
and 
are relatively prime positive integers. Find 
.
Solution
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
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 | ||
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