[$\binom{m}{n}$ below means the number of ways $n$ objects can be chosen from $m$ objects]
Selecting Cricket Team and Captain
The problem involves two main selection processes:
- Choosing a cricket team of 11 players from 40 players.
- Choosing a captain from the selected 11 players.
We need to find the total number of ways both actions can be performed.
Ways to Choose the Cricket Team
The number of ways to choose 11 players from a group of 40 is calculated using combinations, denoted as $\binom{m}{n}$. Here, $m=40$ and $n=11$.
Number of ways to choose the team = $\binom{40}{11}$
Ways to Choose the Captain
Once the team of 11 players is chosen, one player needs to be selected as the captain. There are 11 players in the team.
Number of ways to choose the captain from 11 players = $\binom{11}{1} = 11$
Total Number of Ways
To find the total number of ways to perform both actions (selecting the team AND selecting the captain), we use the multiplication principle.
Total Ways = (Ways to choose the team) $\times$ (Ways to choose the captain)
Total Ways = $\binom{40}{11} \times 11$
Total Ways = $11\binom{40}{11}$
Conclusion
Therefore, the total number of ways to choose an 11-player cricket team from 40 players and then select a captain from the chosen team is $11\binom{40}{11}$.
Important Questions from Permutation and Combination (Notes)
- In how many ways can 10 men be divided into two groups of 4 men and 6 men?
- Out of 5 consonants and 4 vowels, how many words of 3 consonants and 3 vowels can be made?
- How many 5-digit numbers can be formed from the digits 0, 2, 3, 4, 6, 7 and 9, using each at most once, which are divisible by 5?
- In how many distinguishable ways can the letters of the word CHANCE be arranged?
- The maximum number of points formed by intersection of all pairs of diagonals of convex octagon is
