In a cricket against Zimbabwa, Azhar wants to bat before jabeja and wants to bat before Ganguli. Number of possible batting orders with the above restrictions, if the remaining eight team members are prepared to bat any given place, is

Options

A. `(11!)/(8!)`
B. `(11!)/(3)`
C. `(11!)/(3!)`
D. `(11!)/(2!)`

Explanation

To solve the problem, we need to determine the number of possible batting orders for the cricket team with the given restrictions. Let's break down the solution step by step.

### Step 1: Understand the restrictions
We have three players: Azhar (A), Zadeja (Z), and Ganguly (G). The restrictions are:
- Azhar must bat before Zadeja (A < Z)
- Zadeja must bat before Ganguly (Z < G)

This means the order of these three players must always be A, Z, G.

### Step 2: Choose positions for A, Z, and G
Since there are 11 players in total, we need to choose 3 positions out of the 11 for A, Z, and G. The number of ways to choose 3 positions from 11 is given by the combination formula:

\[ \text{Number of ways to choose 3 positions} = \binom{11}{3} = \frac{11!}{3!(11-3)!} = \frac{11!}{3! \cdot 8!} \]

### Step 3: Arrange the remaining players
After placing A, Z, and G in their chosen positions, we have 8 remaining players. These players can occupy the remaining 8 positions in any order. The number of ways to arrange these 8 players is given by:

\[ \text{Number of arrangements of remaining players} = 8! \]

### Step 4: Calculate the total arrangements
The total number of batting orders considering the restrictions is the product of the number of ways to choose positions for A, Z, and G and the arrangements of the remaining players:

\[ \text{Total arrangements} = \binom{11}{3} \cdot 8! = \frac{11!}{3! \cdot 8!} \cdot 8! \]

The \(8!\) in the numerator and denominator cancels out:

\[ \text{Total arrangements} = \frac{11!}{3!} \]

### Step 5: Calculate the final answer
Now we can compute the final answer:

\[ 3! = 6 \]
\[ 11! = 39916800 \]
\[ \text{Total arrangements} = \frac{39916800}{6} = 6652800 \]

Thus, the total number of possible batting orders with the given restrictions is **6652800**.