find the value of `126!-125!`

Answer

To find the value of \( 126! - 125! \), we can use the property of factorials. Here's the step-by-step solution:

### Step 1: Understand the factorial notation
The factorial of a number \( n \), denoted as \( n! \), is the product of all positive integers from 1 to \( n \). For example:
- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

### Step 2: Express \( 126! \) in terms of \( 125! \)
Using the property of factorials, we can express \( 126! \) as:
\[ 126! = 126 \times 125! \]

### Step 3: Substitute \( 126! \) in the expression
Now we can substitute \( 126! \) in the expression \( 126! - 125! \):
\[ 126! - 125! = (126 \times 125!) - 125! \]

### Step 4: Factor out \( 125! \)
Next, we can factor out \( 125! \) from the expression:
\[ 126! - 125! = 125! \times (126 - 1) \]

### Step 5: Simplify the expression
Now simplify the expression inside the parentheses:
\[ 126 - 1 = 125 \]
So we have:
\[ 126! - 125! = 125! \times 125 \]

### Step 6: Write the final answer
Thus, the value of \( 126! - 125! \) is:
\[ 125! \times 125 \]