If n is a positive integer, then `((n+4)!)/((n+7)!)=`

Options

A. `(1)/((n+1)(n+2)(n+3))`
B. `(1)/(n+5)`
C. `(1)/((n+5)(n+6))`
D. `(1)/((n+5)(n+6)(n+7))`

Explanation

To solve the expression \(\frac{(n+4)!}{(n+7)!}\), we can follow these steps:

### Step 1: Write the factorials in expanded form
We know that the factorial of a number \(k\) is defined as:
\[ k! = k \times (k-1) \times (k-2) \times \ldots \times 3 \times 2 \times 1 \]
Thus, we can express \((n+7)!\) in terms of \((n+4)!\):
\[ (n+7)! = (n+7) \times (n+6) \times (n+5) \times (n+4)! \]

### Step 2: Substitute into the original expression
Now, we can substitute this expression for \((n+7)!\) into our original fraction:
\[ \frac{(n+4)!}{(n+7)!} = \frac{(n+4)!}{(n+7) \times (n+6) \times (n+5) \times (n+4)!} \]

### Step 3: Simplify the expression
Notice that \((n+4)!\) appears in both the numerator and the denominator, so we can cancel it out:
\[ \frac{(n+4)!}{(n+7) \times (n+6) \times (n+5) \times (n+4)!} = \frac{1}{(n+7) \times (n+6) \times (n+5)} \]

### Final Result
Thus, the simplified expression is:
\[ \frac{1}{(n+5)(n+6)(n+7)} \]