If `(1+x)^(15)+C_(0)+C_(1)x+C_(2)x^(2)+C_(3)x^(3)+ . . .+C_(15)x^(15) and (k=C_(2)+2C_(3)+3C_(4)+ . . .+14C_(15))` then the value of `(k-993)/(1000)` is equal to______

Answer

To solve the problem, we need to find the value of \( k \) defined as:

\[ k = C_2 + 2C_3 + 3C_4 + \ldots + 14C_{15} \]

where \( C_n \) represents the binomial coefficients from the expansion of \( (1+x)^{15} \).

### Step 1: Understanding the Binomial Expansion
The binomial expansion of \( (1+x)^{15} \) is given by:

\[ (1+x)^{15} = C_0 + C_1 x + C_2 x^2 + C_3 x^3 + \ldots + C_{15} x^{15} \]

### Step 2: Finding the Value of \( k \)
To find \( k \), we can use the property of binomial coefficients. We can express \( k \) in terms of derivatives of \( (1+x)^{15} \):

\[ k = \sum_{n=2}^{15} (n-1) C_n = \sum_{n=2}^{15} n C_n - \sum_{n=2}^{15} C_n \]

The first sum can be computed using the derivative of \( (1+x)^{15} \):

\[ \frac{d}{dx} (1+x)^{15} = 15(1+x)^{14} \]

Evaluating this at \( x=1 \):

\[ 15(1+1)^{14} = 15 \cdot 2^{14} \]

### Step 3: Evaluating the Second Sum
The second sum \( \sum_{n=0}^{15} C_n \) is simply \( (1+1)^{15} = 2^{15} \).

Thus, we have:

\[ k = 15 \cdot 2^{14} - 2^{15} \]

### Step 4: Simplifying \( k \)
We can simplify \( k \):

\[ k = 15 \cdot 2^{14} - 2 \cdot 2^{14} = (15-2) \cdot 2^{14} = 13 \cdot 2^{14} \]

### Step 5: Calculating \( 2^{14} \)
Calculating \( 2^{14} \):

\[ 2^{14} = 16384 \]

Thus,

\[ k = 13 \cdot 16384 = 212992 \]

### Step 6: Finding \( \frac{k - 993}{1000} \)
Now we need to calculate:

\[ \frac{k - 993}{1000} = \frac{212992 - 993}{1000} = \frac{211999}{1000} = 211.999 \]

Since we are looking for an integer value, we take the floor value:

\[ \frac{211999}{1000} = 211 \]

### Final Answer
The final value is:

\[ \boxed{211} \]