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  • A cricket team consists of 20 players, including 5 batsmen, 8 bowlers, and 7 wicketkeepers. Determine the number of ways that a team of 13 players can be chosen from this group in order to include

A cricket team consists of 20 players, including 5 batsmen, 8 bowlers, and 7 wicketkeepers. Determine the number of ways that a team of 13 players can be chosen from this group in order to include at least 4 bowlers, 3 batsmen, and 2 wicketkeepers.

Option: 1

24010


Option: 2

22102


Option: 3

21030

 


Option: 4

23410


Answers (1)

best_answer

There are 20 in a cricket team.

There are 5 batsmen, 8 bowlers, and 7 wicketkeepers.

Case 1: The team consists of 4 bowlers, 4 batsmen, and 5 wicketkeepers 

\begin{aligned} & { }^8 C_4 \times{ }^5 C_4 \times{ }^7 C_5=\frac{8 !}{4 ! 4 !} \times \frac{5 !}{4 ! 1 !} \times \frac{7 !}{5 ! 2 !} \\ & { }^8 C_4 \times{ }^5 C_4 \times{ }^7 C_5=70 \times 5 \times 21 \\ & { }^8 C_4 \times{ }^5 C_4 \times{ }^7 C_5=7350 \end{aligned}

Case 2: The team consists of 5 bowlers, 3 batsmen, and 5 wicketkeepers 

\begin{aligned} & { }^8 C_5 \times{ }^5 C_3 \times{ }^7 C_5=\frac{8 !}{5 ! 3 !} \times \frac{5 !}{3 ! 2 !} \times \frac{7 !}{5 ! 2 !} \\ & { }^8 C_5 \times{ }^5 C_3 \times{ }^7 C_5=56 \times 10 \times 21 \\ & { }^8 C_5 \times{ }^5 C_3 \times{ }^7 C_5=11760 \end{aligned}

Case 3: The team consists of 4 bowlers, 3 batsmen, and 6 wicketkeepers 

\begin{aligned} & { }^8 C_4 \times{ }^5 C_3 \times{ }^7 C_6=\frac{8 !}{4 ! 4 !} \times \frac{5 !}{3 ! 2 !} \times \frac{7 !}{6 ! 1 !} \\ & { }^8 C_4 \times{ }^5 C_3 \times{ }^7 C_6=70 \times 10 \times 7 \\ & { }^8 C_4 \times{ }^5 C_3 \times{ }^7 C_6=4900 \end{aligned}

Therefore, the total number of ways is 7350+11760+4900 = 24010 ways.

 

Posted by

manish painkra

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