Key Concepts and Formulas

• Stars and Bars (Non-negative Solutions): The number of non-negative integer solutions to $x_1 + x_2 + \dots + x_k = n$ is ${}^{n+k-1}{C_{k-1}}$.
• Stars and Bars (Positive Solutions): The number of positive integer solutions to $x_1 + x_2 + \dots + x_k = n$ is ${}^{n-1}{C_{k-1}}$.
• Combinations: The number of ways to choose $k$ items from a set of $n$ distinct items is ${}^{n}{C_k} = \frac{n!}{k!(n-k)!}$.

Step-by-Step Solution

Statement 1 Analysis

Step 1: Define the problem mathematically. We want to distribute 10 identical balls into 4 distinct boxes, such that each box has at least one ball. Let $x_i$ be the number of balls in box $i$, where $i = 1, 2, 3, 4$. We want to find the number of integer solutions to: $x_1 + x_2 + x_3 + x_4 = 10$ with the constraint $x_i \ge 1$ for all $i$.

Step 2: Transform the variables to allow for non-negative solutions. Since each $x_i$ must be at least 1, let $y_i = x_i - 1$. Then $y_i \ge 0$. Substituting into the equation: $(y_1 + 1) + (y_2 + 1) + (y_3 + 1) + (y_4 + 1) = 10$ $y_1 + y_2 + y_3 + y_4 = 10 - 4 = 6$ Now we want to find the number of non-negative integer solutions to this new equation.

Step 3: Apply the Stars and Bars formula. We have $n = 6$ (the sum) and $k = 4$ (the number of variables). The number of non-negative integer solutions is: ${}^{n+k-1}{C_{k-1}} = {}^{6+4-1}{C_{4-1}} = {}^{9}{C_3}$ Therefore, Statement 1 is true.

Statement 2 Analysis

Step 1: Define the problem. We want to choose 3 places from 9 distinct places. This is a combination problem because the order in which we choose the places does not matter.

Step 2: Apply the combination formula. The number of ways to choose 3 places from 9 is: ${}^{9}{C_3} = \frac{9!}{3!(9-3)!} = \frac{9!}{3!6!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 3 \times 4 \times 7 = 84$

Step 3: Conclusion for Statement 2. The number of ways of choosing any 3 places from 9 different places is ${}^9{C_3}$. Therefore, Statement 2 is true.

Relationship between Statements

Both statements are true, and both evaluate to ${}^9{C_3}$. However, Statement 2 does not explain Statement 1. Statement 1 is about distributing identical objects into distinct boxes with a minimum requirement. The solution uses a transformation and the Stars and Bars method. Statement 2 is a simple combination problem. While the results are numerically equal, the underlying principles and contexts are different. Statement 2 does not provide any insight into the logic behind distributing identical balls into boxes.

Common Mistakes & Tips

• Distinguish between identical and distinct objects/boxes. Stars and Bars applies to identical objects into distinct boxes.
• Remember to transform variables when there are minimum requirements. Let $y_i = x_i - m$ to ensure $y_i \ge 0$.
• Numerical equality does not imply explanation. Understand the underlying principles behind each statement.

Summary

Statement 1 is true, as the number of ways to distribute 10 identical balls into 4 distinct boxes such that no box is empty is ${}^9{C_3}$. Statement 2 is also true, as the number of ways to choose 3 places from 9 different places is ${}^9{C_3}$. However, Statement 2 is not a correct explanation for Statement 1, as they represent different combinatorial problems with distinct underlying principles. This corresponds to option (A).

The final answer is \boxed{A}.