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 3 places
from $9$different place is ${ }^9{C}_3$

Let $x_i=$ number of balls put in $ith$box,

then, $x_1+x_2+x_3+x_4=10$ where, $x_i\ge 1$

Put $x_i=y_i+1$ so that equation becomes

$y_1+y_2+y_3+y_4=6$. where $y_i\ge 0$

Number of non-negative integral solution of the above equation
= Number of ways of arranging $6$ identical
balls and 3 identical separators

$=\frac{(6+3)!}{6!3!}{=}^9{C}_3$ number of ways of choosing$3$
places out of$9$ different places.