Question

The chance that $13$ card combination from a pack of $52$ playing cards is dealt to a player in a game of bridge, in which $9$ cards are of the same suit, is
A. $\dfrac{{4 \cdot {}^{13}{C_9} \cdot {}^{39}{C_4}}}{{{}^{52}{C_{13}}}}$
B. $\dfrac{{4! \cdot {}^{13}{C_9} \cdot {}^{39}{C_4}}}{{{}^{52}{C_{13}}}}$
C. $\dfrac{{{}^{13}{C_9} \cdot {}^{39}{C_4}}}{{{}^{52}{C_{13}}}}$
D. $2\dfrac{{{}^{13}{C_9} \cdot {}^{39}{C_4}}}{{{}^{52}{C_{13}}}}$

Answer 1

Hint: 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 $13$ card combinations from a pack of $52$ 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 $13$ card combination from a pack of $52$ playing cards is dealt with by a player in a game of bridge, in which $9$ cards are of the same suit.

Complete step by step answer:
We need to find the chance that $13$ a card combination from a pack of $52$ playing cards is dealt with by a player in a game of bridge, in which $9$ cards are of the same suit.
Each player needs to choose a suit from four suits.
That is, each player has ${}^4{C_1}$ ways.
Also, the player selects $9$ cards that are of the same suit, that is ${}^{13}{C_9}$ .
And, he needs to select four cards from the remaining $39$ cards, that is ${}^{39}{C_4}$ .
We know that the total number of ways is ${}^{52}{C_{13}}$ .
Hence, the required probability is$\dfrac{{{}^4{C_1} \cdot {}^{13}{C_9} \cdot {}^{39}{C_4}}}{{{}^{52}{C_{13}}}}$
That is, $\dfrac{{4 \cdot {}^{13}{C_9} \cdot {}^{39}{C_4}}}{{{}^{52}{C_{13}}}}$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 $13$ card combinations from a pack of $52$ playing cards to each player in a clockwise rotation around the table.