`(7!) / (5!)` is :

Options

A. 7!
B. 2!
C. 42
D. 24

Answer

To solve the expression \((7!) / (5!)\), we can follow these steps:

### Step 1: Understand Factorials
Factorial notation \(n!\) means the product of all positive integers from 1 to \(n\). Therefore:
- \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\)
- \(5! = 5 \times 4 \times 3 \times 2 \times 1\)

### Step 2: Rewrite \(7!\)
We can express \(7!\) in terms of \(5!\):
\[ 7! = 7 \times 6 \times 5! \]

### Step 3: Substitute into the Expression
Now, substitute \(7!\) in the original expression:
\[ \frac{7!}{5!} = \frac{7 \times 6 \times 5!}{5!} \]

### Step 4: Cancel \(5!\)
Since \(5!\) appears in both the numerator and the denominator, we can cancel it out:
\[ \frac{7 \times 6 \times 5!}{5!} = 7 \times 6 \]

### Step 5: Calculate the Result
Now, calculate \(7 \times 6\):
\[ 7 \times 6 = 42 \]

### Final Answer
Thus, \(\frac{7!}{5!} = 42\).

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