Question

What is the number of ways two players can get a full house?

Answer 1

Deck Of Cards

A deck of 52 cards has 4 suits, Spades, Clubs, Diamonds and Hearts

There are 13 cards in each suit, from 2 till 10, the 3 face cards J, Q, K and finally Ace.

Therefore there are 4 cards of each type i.e 4 kings, 4 queens etc

A full house is when a player gets a 3 of a kind of 1 type and another 2 of a kind of another type.

1st Player: for the 3 of a kind, we will choose from any 13 cards in 13C1 = 13 ways (since there are 13 types of cards).

Then we choose 3 cards from this in 4C3 = 4 ways (Since each card has 4 types)

Now from the remaining 12 we will choose 1 card for the 2 of a kind in 12C1 = 12 ways

From this we will choose any 2 in 4C2 = 6 ways

Total ways for player 1 = 13 * 4 * 1 2 * 6 = 3744

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2nd Player: for the 3 of a kind, we will choose from any 11 cards in 11C1 = 11 ways (since 2 types have gone to Player 1).

Then we choose 3 cards from this in 4C3 = 4 ways (Since each card has 4 types)

For the 2 of a kind, there are 2 possibilities

(a) From the remaining 10 types we will choose 1 card for the 2 of a kind in 10C1 = 10 ways

From this we will choose any 2 in 4C2 = 6 ways

(b) The remaining 2 cards which player got for 2 of a kind will be received by player 2 in 2C2 = 1 way

The total ways for choosing 2 of a kind = 10 * (6 + 1) = 70

Total ways for player 2 = 11 * 4 * 70 = 3080

The total ways for player 1 and player 2 to get a full house = 3744 * 3080 = 11,531,520