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 
of 
cards sharing the same number and of them sharing same color.
pairs of cards can't be the same or else there will be card which are completely same.
WLOG the numbers are 
and and the colors are 
and 
Then we can get cases:
Case One:
and 
in this case, we can discard 
.
there are 
situations in this case.
Case Two:
and
In this case, we can't discard.
There are 
situations in this case
So the probability is 
The answer is 
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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