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Question: The chance that \(13\) card combination from a pack of \(52\) playing cards is dealt to a player in ...

The chance that 1313 card combination from a pack of 5252 playing cards is dealt to a player in a game of bridge, in which 99 cards are of the same suit, is
A. 4⋅13C9⋅39C452C13\dfrac{{4 \cdot {}^{13}{C_9} \cdot {}^{39}{C_4}}}{{{}^{52}{C_{13}}}}
B. 4!⋅13C9⋅39C452C13\dfrac{{4! \cdot {}^{13}{C_9} \cdot {}^{39}{C_4}}}{{{}^{52}{C_{13}}}}
C. 13C9⋅39C452C13\dfrac{{{}^{13}{C_9} \cdot {}^{39}{C_4}}}{{{}^{52}{C_{13}}}}
D. 213C9⋅39C452C132\dfrac{{{}^{13}{C_9} \cdot {}^{39}{C_4}}}{{{}^{52}{C_{13}}}}

Explanation

Solution

Generally, a game of bridge is played by four players. Here, the first player and third player are made as partners. Similarly, the second player and the fourth player will be partners. Therefore, a game of bridge consists of two pairs of partner players at a card table. Now, one of the players deals 1313 card combinations from a pack of 5252 playing cards to each player in a clockwise rotation around the table. Each player has to play with a trick to win the game. There are four suits in the bridge; they are ranked as spades (highest), hearts, diamonds, and clubs (lowest).
In a game of bridge, probability plays a vital role in making strategies. Here, our question is in the game of bridge, where we need to find the chance that a 1313 card combination from a pack of 5252 playing cards is dealt with by a player in a game of bridge, in which 99 cards are of the same suit.

Complete step by step answer:
We need to find the chance that 1313 a card combination from a pack of 5252 playing cards is dealt with by a player in a game of bridge, in which 99 cards are of the same suit.
Each player needs to choose a suit from four suits.
That is, each player has 4C1{}^4{C_1} ways.
Also, the player selects 99 cards that are of the same suit, that is 13C9{}^{13}{C_9} .
And, he needs to select four cards from the remaining 3939 cards, that is 39C4{}^{39}{C_4} .
We know that the total number of ways is 52C13{}^{52}{C_{13}} .
Hence, the required probability is4C1⋅13C9⋅39C452C13\dfrac{{{}^4{C_1} \cdot {}^{13}{C_9} \cdot {}^{39}{C_4}}}{{{}^{52}{C_{13}}}}
That is, 4⋅13C9⋅39C452C13\dfrac{{4 \cdot {}^{13}{C_9} \cdot {}^{39}{C_4}}}{{{}^{52}{C_{13}}}}is the required answer.

So, the correct answer is “Option A”.

Note: A game of bridge consists of two pairs of partner players at a card table. Now, one of the players deals 1313 card combinations from a pack of 5252 playing cards to each player in a clockwise rotation around the table.