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 | ||
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