The value of ` [ ( 2009! + 2006!)/(2008! + 2007!)] ` is `K`. Then value of `K/1004`
(` [ *] ` denotes greatest integer function.)

Options

A. 3
B. 2
C. 4
D. 1

Answer

To solve the problem, we need to evaluate the expression:

\[ K = \frac{2009! + 2006!}{2008! + 2007!} \]

### Step 1: Factor out common terms in the numerator and denominator
In the numerator, we can factor out \(2006!\):

\[ 2009! + 2006! = 2006!(2009 \times 2008 + 1) \]

In the denominator, we can factor out \(2007!\):

\[ 2008! + 2007! = 2007!(2008 + 1) = 2007! \times 2009 \]

### Step 2: Rewrite the expression for K
Now we can rewrite \(K\) as:

\[ K = \frac{2006!(2009 \times 2008 + 1)}{2007! \times 2009} \]

### Step 3: Simplify the expression
Since \(2007! = 2007 \times 2006!\), we can substitute this into the expression for \(K\):

\[ K = \frac{2006!(2009 \times 2008 + 1)}{2007 \times 2006! \times 2009} \]

The \(2006!\) cancels out:

\[ K = \frac{2009 \times 2008 + 1}{2007 \times 2009} \]

### Step 4: Further simplify K
Now we can simplify \(K\):

\[ K = \frac{2009 \times 2008}{2007 \times 2009} + \frac{1}{2007 \times 2009} \]

The \(2009\) cancels out:

\[ K = \frac{2008}{2007} + \frac{1}{2007 \times 2009} \]

### Step 5: Calculate the value of K
The first term simplifies to:

\[ \frac{2008}{2007} \approx 1.0005 \]

The second term \(\frac{1}{2007 \times 2009}\) is a very small number, so we can approximate \(K\) as:

\[ K \approx 1.0005 \]

### Step 6: Find the greatest integer function of K
Since \(K\) is slightly more than 1, the greatest integer function \([K]\) is:

\[ [K] = 1 \]

### Step 7: Calculate \(\frac{K}{1004}\)
Now we need to find:

\[ \frac{K}{1004} = \frac{1}{1004} \approx 0.000996 \]

### Step 8: Apply the greatest integer function
Since \(\frac{K}{1004}\) is less than 1, the greatest integer function is:

\[ \left[\frac{K}{1004}\right] = 0 \]

### Final Result
Thus, the value of \(\left[\frac{K}{1004}\right]\) is:

\[ \boxed{0} \]