Let $ n$, $ k$, $ p$ be positive integers with $ 2 \le k \le \frac {n}{p \plus{} 1}$. Let $ n$ distinct points on a circle be given. These points are colored blue and red so that exactly $ k$ points are blue and, on each arc determined by two consecutive blue points in clockwise direction, there are at least $ p$ red points. How many such colorings are there?
1. **Placing the Blue Points:**
   - We start by placing the \( k \) blue points anywhere on the circle. Since the points are distinct, we can choose any \( k \) out of the \( n \) points to be blue. 

2. **Placing the Necessary Red Points:**
   - Between each pair of consecutive blue points, we need to place at least \( p \) red points. Since there are \( k \) pairs of consecutive blue points, we need at least \( kp \) red points in total to satisfy this condition.

3. **Counting Remaining Points:**
   - After placing the \( k \) blue points and the necessary \( kp \) red points, we have \( n - k - kp \) points left. These remaining points must all be colored red.

4. **Distributing Remaining Red Points:**
   - The distinctness of the coloring depends on how many of the remaining red points are placed between each pair of consecutive blue points. Let \( p_i \) be the number of remaining red points between the \( i \)-th and \((i+1)\)-th blue points (excluding the necessary \( p \) red points). We need to solve for the number of nonnegative integer solutions to the equation:
     \[
     \sum_{i=1}^k p_i = n - k - kp
     \]
   - The number of nonnegative integer solutions to this equation is given by the stars and bars theorem, which states that the number of solutions is:
     \[
     \binom{(n - k - kp) + (k - 1)}{k - 1} = \binom{n - kp - 1}{k - 1}
     \]

5. **Adjusting for Symmetry:**
   - Since the circle is symmetric, we have overcounted the number of distinct colorings by a factor of \( k \) (because rotating the circle by any number of positions would result in the same coloring). Therefore, we need to divide by \( k \) to account for these symmetries.

6. **Final Calculation:**
   - The total number of distinct colorings is:
     \[
     \frac{1}{k} \binom{n - kp - 1}{k - 1}
     \]

The final answer is \(\boxed{\frac{1}{k} \binom{n - kp - 1}{k - 1}}\)