Let n and k be positive integers such that `n gt (k(k+1))/2`. The number of solutions `(x_(1),x_(2), . ..x_(k)),x_(1) ge 1, x_(2) ge 2,, . . .x_(k) ge k` for all integers satisfying `x_(1)+x_(2)+ . . .+x_(k)=n` is:

Answer

To solve the problem, we need to find the number of solutions to the equation:

\[ x_1 + x_2 + \ldots + x_k = n \]

with the constraints:

- \( x_1 \geq 1 \)
- \( x_2 \geq 2 \)
- \( x_3 \geq 3 \)
- ...
- \( x_k \geq k \)

### Step 1: Transform the Variables
To simplify the problem, we can perform a change of variables. Let:

- \( y_1 = x_1 - 1 \) (thus \( y_1 \geq 0 \))
- \( y_2 = x_2 - 2 \) (thus \( y_2 \geq 0 \))
- \( y_3 = x_3 - 3 \) (thus \( y_3 \geq 0 \))
- ...
- \( y_k = x_k - k \) (thus \( y_k \geq 0 \))

Now, substituting these into the original equation gives us:

\[ (y_1 + 1) + (y_2 + 2) + (y_3 + 3) + \ldots + (y_k + k) = n \]

This simplifies to:

\[ y_1 + y_2 + y_3 + \ldots + y_k + \frac{k(k + 1)}{2} = n \]

### Step 2: Rearranging the Equation
Rearranging this equation, we find:

\[ y_1 + y_2 + y_3 + \ldots + y_k = n - \frac{k(k + 1)}{2} \]

Let:

\[ m = n - \frac{k(k + 1)}{2} \]

### Step 3: Finding Non-negative Integer Solutions
Now, we need to find the number of non-negative integer solutions to the equation:

\[ y_1 + y_2 + y_3 + \ldots + y_k = m \]

The number of non-negative integer solutions to the equation \( y_1 + y_2 + \ldots + y_k = m \) can be found using the "stars and bars" theorem, which states that the number of solutions is given by:

\[ \binom{m + k - 1}{k - 1} \]

### Step 4: Final Expression
Substituting \( m \) back into the formula gives us:

\[ \text{Number of solutions} = \binom{n - \frac{k(k + 1)}{2} + k - 1}{k - 1} \]

### Conclusion
Thus, the number of solutions \( (x_1, x_2, \ldots, x_k) \) satisfying the given conditions is:

\[ \binom{n - \frac{k(k + 1)}{2} + k - 1}{k - 1} \]