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

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 $4+9=\boxed{013}$

2017 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
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2017 AIME II Problems/Problem 9

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