Dominion starting hand

Discrete Mathematics Level 3

In the game Dominion, each player begins with a shuffled deck of 10 cards, consisting of 7 Coppers and 3 Estates. In each turn, a player draws 5 cards, and at the end of the turn, a player discards all cards in play; thus, there is no card retained between turns. Copper and Estate don't allow the player to draw more cards, either, so a player's first two turns are always composed of drawing all the 10 starting cards in some order. For example, a player can draw 4 Coppers and 1 Estate in the first turn, then the remaining 3 Coppers and 2 Estates in the next turn.

An opening is the number of Coppers in hand for the first two turns. For the above example, the opening is called a 4/3 opening: 4 Coppers are drawn in the first turn, then 3 Coppers in the second. A 3/4 opening is functionally identical to 4/3, and likewise 2/5 is identical to 5/2. It can also be easily shown that there is no other kind of opening, so there are only two distinct openings: 5/2 or 4/3. Which opening is obtained depends wholly on the luck of the first shuffle.

If the probability of getting a 5/2 opening is \(\frac{a}{b}\) where \(a,b\) are positive integers that are relatively prime, determine the value of \(a+b\).

Assumption: Count both 5/2 and 2/5 opening, since they are identical as stated.


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