1. Check Statement 1

We need the number of ways to distribute $10$ identical balls into $4$ distinct boxes such that no box is empty.

This is the number of positive integer solutions of $x_1+x_2+x_3+x_4=10,$ where $x_i\ge 1$ for all $i$.

Using the stars and bars formula, the number of positive integer solutions of $x_1+x_2+\cdots+x_r=n$ is $\binom{n-1}{r-1}.$

Here, $n=10$ and $r=4$, so $\text{Number of ways}=\binom{10-1}{4-1}=\binom{9}{3}.$

So, Statement 1 is true.

2. Check Statement 2

Statement 2 says: the number of ways of choosing any $3$ places from $9$ different places is $\binom{9}{3}.$

This is exactly the definition of combinations.

Hence, Statement 2 is true.

3. Does Statement 2 explain Statement 1?

Yes.

In the stars and bars method, distributing $10$ identical balls into $4$ distinct non-empty boxes can be visualized as:

• Write $10$ stars in a row.
• To divide them into $4$ non-empty groups, place $3$ separators among the $9$ gaps between consecutive stars.

Example structure: $*\ *\ |\ *\ *\ *\ |\ *\ |\ *\ *\ *\ *$

There are exactly $9$ available gaps between $10$ stars, and we must choose $3$ of them for separators. Thus the number of distributions is $\binom{9}{3}.$

So Statement 2 gives the combinatorial reasoning behind Statement 1.

Hence, Statement 2 is a correct explanation for Statement 1.

4. Conclusion

• Statement 1: True
• Statement 2: True
• Statement 2 correctly explains Statement 1

Therefore, the correct option is D.

5. Comparison with stored answer

Stored correct answer: A

But our derivation shows the correct answer should be D, because Statement 2 is indeed the standard explanation of Statement 1 via stars and bars.