Statement-1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is `""^(9)C_(3)`.
Statement-2: The number of ways of choosing any 3 places from 9 different places is `""^(9)C_(3)`.

Options

A. Statement-1 is true, statement-2 is true, statement-2 is a correct explanation for statement-1
B. Statement-1 is true, statement-2 is true, statement-2 is not a correct explanation for statement-1
C. Statement-1 is true, statement-2 is false
D. Statement-1 is false, statement-2 is true

Explanation

To solve the given problem, we will analyze both statements step by step.

### Step 1: Understanding Statement 1
We need to distribute 10 identical balls into 4 distinct boxes such that no box is empty.

1. **Identifying the Problem**: Since no box can be empty, we first place 1 ball in each of the 4 boxes. This ensures that all boxes have at least one ball.

2. **Calculating Remaining Balls**: After placing 1 ball in each box, we have used 4 balls. Therefore, the number of balls left to distribute is:
\[ 10 - 4 = 6 \text{ balls} \]

3. **Using the Stars and Bars Theorem**: Now, we need to distribute these 6 identical balls into 4 distinct boxes. The problem can be transformed into finding the number of non-negative integer solutions to the equation:
\[ x_1 + x_2 + x_3 + x_4 = 6 \]
where \(x_i\) represents the number of additional balls in box \(i\).

4. **Applying the Formula**: According to the Stars and Bars theorem, the number of ways to distribute \(n\) identical items into \(r\) distinct groups is given by:
\[ \binom{n + r - 1}{r - 1} \]
Here, \(n = 6\) (remaining balls) and \(r = 4\) (boxes). Thus, we have:
\[ \text{Number of ways} = \binom{6 + 4 - 1}{4 - 1} = \binom{9}{3} \]

### Conclusion for Statement 1:
The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is indeed \(\binom{9}{3}\).

### Step 2: Understanding Statement 2
We need to find the number of ways to choose any 3 places from 9 different places.

1. **Identifying the Problem**: This is a straightforward combination problem where we are selecting 3 places out of 9.

2. **Using the Combination Formula**: The number of ways to choose \(r\) items from \(n\) items is given by:
\[ \binom{n}{r} \]
In our case, \(n = 9\) and \(r = 3\). Therefore, the number of ways is:
\[ \binom{9}{3} \]

### Conclusion for Statement 2:
The number of ways of choosing any 3 places from 9 different places is also \(\binom{9}{3}\).

### Final Conclusion:
Both statements are true:
- Statement 1 is true as it correctly applies the Stars and Bars theorem.
- Statement 2 is true as it correctly uses the combination formula.