When Agnishom asked the prettiest girl in his class to go on a date with him, she pulled out a standard deck of 52 cards.

She said, "You have to choose 2 cards from this pack without replacement. If both the cards belong to the heart suit, we are on."

As she started shuffling the deck, unknown to either of them, 3 cards inadvertently fell out of the pack at random. In effect the deck was consisting of 49 cards when Agnishom chose his two cards.

Given this scenario, the probability that Agnishom goes on the date i.e. that he draws two hearts without replacement can be expressed as $\dfrac{a}{b}$ where $a$ and $b$ are co-prime positive integers.

Find the value of $a+b$ .

The answer is 18.

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First, we will have to realize that this is just the same thing as choosing 5 random cards with trying to have the last 2 cards heart cards.

To do this, we first notice that the last 2 heart cards can be chosen in $13 \times 12$ ways.

We have every freedom in choosing the first three cards except that we cannot choose the two cards we fixed for the choice of the last two. So, we have $50 \times 49 \times 48$ ways to choose the first three cards.

Thus, there are a total of $13 \times 12 \times 50 \times 49 \times 48$ ways he can get lucky.

Now, in how many ways can the cards be chosen

anyway? It is obviously ${ 52 \choose 5 } 5!$So, $\text{P(Agnishom goes to the date)} = \frac{13 \times 12 \times 50 \times 49 \times 48 }{{ 52 \choose 5 } 5!}$

Thanks for the problem, Satyen! I'll definitely show her :)