A cricket team of 11 players is to be selected from 16 players including 5 bowlers and 2 wicketkeepers. In how many ways can a team be selected so as to consist of exactly 3 bowlers and 1 wicketkeeper?

Options

A. b. 720
B. b. 272
C. c. 850
D. d. none of (a),(b) ,(c )

Answer

To solve the problem of selecting a cricket team of 11 players from 16 players (including 5 bowlers and 2 wicketkeepers) such that the team consists of exactly 3 bowlers and 1 wicketkeeper, we can follow these steps:

### Step 1: Identify the requirements
We need to select:
- 3 bowlers from 5 available bowlers
- 1 wicketkeeper from 2 available wicketkeepers
- The remaining players will be selected from the other players (not bowlers or wicketkeepers)

### Step 2: Calculate the number of ways to select the bowlers
The number of ways to choose 3 bowlers from 5 can be calculated using the combination formula:
\[ \text{C}(n, r) = \frac{n!}{r!(n-r)!} \]
So, for selecting 3 bowlers from 5:
\[ \text{C}(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 3: Calculate the number of ways to select the wicketkeeper
The number of ways to choose 1 wicketkeeper from 2 is:
\[ \text{C}(2, 1) = \frac{2!}{1!(2-1)!} = 2 \]

### Step 4: Calculate the number of remaining players to select
After selecting 3 bowlers and 1 wicketkeeper, we need to select:
\[ 11 - 3 - 1 = 7 \text{ players} \]
These 7 players will be selected from the remaining players. Since we have 16 players in total and have already selected 4 (3 bowlers + 1 wicketkeeper), we have:
\[ 16 - 4 = 12 \text{ remaining players} \]
Out of these 12 remaining players, 5 are bowlers, 1 is a wicketkeeper, and 6 are other players.

### Step 5: Calculate the number of ways to select the remaining players
The number of ways to choose 7 players from the remaining 12 is:
\[ \text{C}(12, 7) = \text{C}(12, 5) = \frac{12!}{5!(12-5)!} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} = 792 \]

### Step 6: Calculate the total number of ways to form the team
Now, we multiply the number of ways to select the bowlers, the wicketkeeper, and the remaining players:
\[ \text{Total ways} = \text{C}(5, 3) \times \text{C}(2, 1) \times \text{C}(12, 7) = 10 \times 2 \times 792 \]
Calculating this gives:
\[ = 10 \times 2 = 20 \]
\[ 20 \times 792 = 15840 \]

### Final Answer
The total number of ways to select the team is **15840**.