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Out of 16 players of cricket team, 4 are bowlers and 2 are wicket keepers. A team of 11 players is to be chosen so as to contain at least 3 bowlers and at least one wicket keeper. The number of ways in which the team be selected is :
- a)2400
- b)2475
- c)2500
- d)none
Correct answer is option 'D'. Can you explain this answer?
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Out of 16 players of cricket team, 4 are bowlers and 2 are wicket keep...
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Out of 16 players of cricket team, 4 are bowlers and 2 are wicket keep...
To solve this problem, we can use the concept of combinations and permutations.
Step 1: Selecting the bowlers
Since there are 4 bowlers in the team, we need to select 3 or 4 bowlers in the team of 11 players.
Case 1: Selecting 3 bowlers
There are 4 bowlers in total, so we need to select 3 out of 4. This can be done in 4C3 ways, which is equal to 4.
Case 2: Selecting 4 bowlers
In this case, we simply select all 4 bowlers, which can be done in 1 way.
Step 2: Selecting the wicket keeper
Since there are 2 wicket keepers in the team, we need to select 1 or 2 wicket keepers in the team of 11 players.
Case 1: Selecting 1 wicket keeper
There are 2 wicket keepers in total, so we need to select 1 out of 2. This can be done in 2C1 ways, which is equal to 2.
Case 2: Selecting 2 wicket keepers
In this case, we simply select both the wicket keepers, which can be done in 1 way.
Step 3: Selecting the remaining players
After selecting the bowlers and the wicket keeper, we need to select the remaining players for the team. Since there are 11 players in total and we have already chosen 3 or 4 bowlers and 1 or 2 wicket keepers, we need to select the remaining players from the remaining pool of players.
The remaining players include the non-bowlers and non-wicket keepers, which is equal to (16 - 4 - 2) = 10 players.
We can select the remaining players in 10C(11-3-1) ways, which is equal to 10C7 = 10.
Step 4: Calculate the total number of ways
To calculate the total number of ways, we need to multiply the number of ways for each case.
Case 1: Selecting 3 bowlers and 1 wicket keeper
Total ways = 4C3 * 2C1 * 10C7 = 4 * 2 * 10 = 80
Case 2: Selecting 4 bowlers and 1 wicket keeper
Total ways = 1 * 2C1 * 10C7 = 2 * 10 = 20
Therefore, the total number of ways to form a team of 11 players with at least 3 bowlers and at least 1 wicket keeper is 80 + 20 = 100.
Hence, the correct answer is none of the given options (D).
Step 1: Selecting the bowlers
Since there are 4 bowlers in the team, we need to select 3 or 4 bowlers in the team of 11 players.
Case 1: Selecting 3 bowlers
There are 4 bowlers in total, so we need to select 3 out of 4. This can be done in 4C3 ways, which is equal to 4.
Case 2: Selecting 4 bowlers
In this case, we simply select all 4 bowlers, which can be done in 1 way.
Step 2: Selecting the wicket keeper
Since there are 2 wicket keepers in the team, we need to select 1 or 2 wicket keepers in the team of 11 players.
Case 1: Selecting 1 wicket keeper
There are 2 wicket keepers in total, so we need to select 1 out of 2. This can be done in 2C1 ways, which is equal to 2.
Case 2: Selecting 2 wicket keepers
In this case, we simply select both the wicket keepers, which can be done in 1 way.
Step 3: Selecting the remaining players
After selecting the bowlers and the wicket keeper, we need to select the remaining players for the team. Since there are 11 players in total and we have already chosen 3 or 4 bowlers and 1 or 2 wicket keepers, we need to select the remaining players from the remaining pool of players.
The remaining players include the non-bowlers and non-wicket keepers, which is equal to (16 - 4 - 2) = 10 players.
We can select the remaining players in 10C(11-3-1) ways, which is equal to 10C7 = 10.
Step 4: Calculate the total number of ways
To calculate the total number of ways, we need to multiply the number of ways for each case.
Case 1: Selecting 3 bowlers and 1 wicket keeper
Total ways = 4C3 * 2C1 * 10C7 = 4 * 2 * 10 = 80
Case 2: Selecting 4 bowlers and 1 wicket keeper
Total ways = 1 * 2C1 * 10C7 = 2 * 10 = 20
Therefore, the total number of ways to form a team of 11 players with at least 3 bowlers and at least 1 wicket keeper is 80 + 20 = 100.
Hence, the correct answer is none of the given options (D).
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