This question has Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. 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 $$^9 C_3$$.

The problem is to compare two statements.

Statement-1: Number of ways of distributing 10 identical balls into 4 distinct boxes, with every box non-empty, equals $$\;{}^{9}C_{3}\;.$$

Statement-2: Number of ways of selecting any 3 places out of 9 different places equals $$\;{}^{9}C_{3}\;.$$

We first verify each statement and then decide whether Statement-2 actually explains Statement-1.

Step 1 - Verifying Statement-1
When identical objects are distributed into distinct boxes with the condition that no box is empty, we use the “stars and bars’’ theorem.
Formula (non-empty): To put $$n$$ identical balls in $$k$$ distinct boxes with each box getting at least one ball, the number of ways is $$ {}^{\,n-1}C_{\,k-1}. $$ Here $$n = 10$$, $$k = 4$$, so $$ {}^{\,10-1}C_{\,4-1}= {}^{9}C_{3}. $$ Thus Statement-1 is true.

Step 2 - Verifying Statement-2
Choosing 3 places out of 9 distinct places is the basic combination problem whose answer is by definition $$ {}^{9}C_{3}. $$ Therefore Statement-2 is also true.

Step 3 - Does Statement-2 explain Statement-1?
In the stars-and-bars argument we picture the 10 balls (stars) lined in a row: ★ ★ ★ … (10 stars). To create 4 non-empty boxes we must insert exactly 3 dividers (bars) into the 9 gaps between consecutive stars:

★ | ★ ★ | ★ ★ ★ | ★ ★ ★ (example)

There are 9 gaps and we must choose any 3 of them to place the bars. So the number of admissible distributions is exactly the number of ways of “choosing 3 places out of 9,” which is $$\;{}^{9}C_{3}\;.$$
Thus Statement-2 not only is true, it supplies the precise reason behind Statement-1.

Hence the correct choice is:
Option D - both statements are true and Statement-2 is a correct explanation for Statement-1.