In how many ways can 7 people line up at a ticket window of a cinema hall?

Answer

To find the number of ways in which 7 people can line up at a ticket window of a cinema hall, we can use the concept of permutations. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We need to determine how many different arrangements can be made with 7 distinct people.

### Step 2: Identify the Formula
The number of ways to arrange \( n \) distinct objects is given by \( n! \) (n factorial). This means we multiply all whole numbers from \( n \) down to 1.

### Step 3: Calculate \( 7! \)
For our case, \( n = 7 \). Therefore, we need to calculate \( 7! \):
\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

### Step 4: Perform the Calculation
Now we will calculate \( 7! \) step by step:
- First, calculate \( 7 \times 6 = 42 \)
- Next, calculate \( 42 \times 5 = 210 \)
- Then, calculate \( 210 \times 4 = 840 \)
- Next, calculate \( 840 \times 3 = 2520 \)
- Then, calculate \( 2520 \times 2 = 5040 \)
- Finally, calculate \( 5040 \times 1 = 5040 \)

So, \( 7! = 5040 \).

### Step 5: Conclusion
Thus, the number of ways in which 7 people can line up at a ticket window is \( 5040 \).