A table has 7 seats, 4 being on one side facing the window and 3 being on the opposite side. In how many ways can 7 people be seated at the table.
If 3 people , X,Y and Z must sit on the side facing the window ?

Answer

To solve the problem of seating 7 people at a table with specific seating arrangements, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the fixed positions**: We know that 3 people (X, Y, and Z) must sit on the side facing the window. Since there are 4 seats on that side, we can choose any 3 of those 4 seats for X, Y, and Z.

2. **Arrange X, Y, and Z**: The 3 people can be arranged among themselves in the 3 chosen seats. The number of ways to arrange 3 people is given by \(3!\) (3 factorial).

\[ 3! = 3 \times 2 \times 1 = 6 \]

3. **Determine the remaining seats**: After seating X, Y, and Z, there will be 1 seat left on the window side and all 3 seats on the opposite side available for the remaining 4 people.

4. **Arrange the remaining people**: The remaining 4 people can occupy the remaining 4 seats (1 on the window side and 3 on the opposite side). The number of ways to arrange these 4 people is given by \(4!\) (4 factorial).

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

5. **Calculate the total arrangements**: The total number of ways to arrange all 7 people at the table, given that X, Y, and Z must sit on the side facing the window, is the product of the arrangements of X, Y, and Z and the arrangements of the remaining 4 people.

\[ \text{Total arrangements} = 3! \times 4! = 6 \times 24 = 144 \]

### Final Answer:
The total number of ways 7 people can be seated at the table, with X, Y, and Z on the side facing the window, is **144**.

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