The image contains a problem about a group of students distributing tokens. Here’s a detailed transcription and breakdown of the problem and solution approach:

Problem Statement: "A group of 5 students receives a total of 10 tokens. The tokens are distributed randomly. Determine the number of ways they can receive the tokens. Assume each student can receive zero or more tokens."

The problem provides the number of students and tokens:

$\text{Students: } 5, \quad \text{Tokens: } 10$

Sub-problem: "Calculate the number of positive integer distributions of different types based on the given data."

Problem Analysis:

This is a typical "stars and bars" combinatorial problem, where we need to determine the number of non-negative integer solutions to the equation:

$x_1 + x_2 + x_3 + x_4 + x_5 = 10$

Where $x_1, x_2, \ldots, x_5$ represent the number of tokens received by each of the 5 students, and $x_i \geq 0$ for all $i$.

Solution:

The number of non-negative integer solutions to the equation above can be calculated using the formula:

$\text{Number of solutions} = \binom{n+k-1}{k-1}$

Where:

• $n$ is the total number of tokens (10),
• $k$ is the number of students (5).

Substituting the values:

$\binom{10+5-1}{5-1} = \binom{14}{4}$

Now, calculate $\binom{14}{4}$:

$\binom{14}{4} = \frac{14 \times 13 \times 12 \times 11}{4 \times 3 \times 2 \times 1} = 1001$

So, there are 1001 ways to distribute 10 tokens among 5 students.

Would you like further details on any specific part of the solution? Here are some related questions:

1. What is the "stars and bars" method in combinatorics?
2. How would the problem change if each student had to receive at least one token?
3. Can you explain the binomial coefficient $\binom{n}{r}$ and its calculation?
4. What are some other combinatorial problems similar to this one?
5. How do you apply combinatorial methods to real-world distribution problems?

Tip: Remember that "stars and bars" is a powerful method for solving distribution problems in combinatorics where order does not matter.