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 expression \( K = \frac{2009! + 2006!}{2008! + 2007!} \), we can simplify it step by step.

### Step 1: Factor out the common terms
We can factor out \( 2006! \) from the numerator and \( 2007! \) from the denominator.

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

### Step 2: Simplify the expression
Now we can simplify \( 2007! \) in the denominator:

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

The \( 2006! \) cancels out:

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

### Step 3: Further simplification
Now we can simplify the expression further:

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

### Step 4: Break down the fraction
This can be broken down into two parts:

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

The first part simplifies to:

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

### Step 5: Evaluate \( K \)
Since \( \frac{1}{2007 \times 2009} \) is a very small number, we can conclude that:

\[ K \approx 2008 + \text{(a very small number)} \]

Thus, the greatest integer function \( \lfloor K \rfloor = 2008 \).

### Step 6: Find \( \frac{K}{1004} \)
Now we need to compute \( \frac{K}{1004} \):

\[ \frac{K}{1004} = \frac{2008}{1004} = 2 \]

### Final Answer
Thus, the value of \( \lfloor \frac{K}{1004} \rfloor \) is \( 2 \).

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