To determine the number of ways to distribute 4 indistinguishable balls into 3 distinguishable boxes, we can use the "stars and bars" theorem from combinatorics.
The number of ways to place \( n \) indistinguishable balls into \( k \) distinguishable boxes can be found by solving the equation:
\[ x_1 + x_2 + x_3 = n \]
where \( x_1, x_2, \) and \( x_3 \) are the numbers of balls in each of the three boxes, respectively, and \( n \) is the total number of balls.
In our case, \( n = 4 \) (the number of balls) and \( k = 3 \) (the number of boxes). The formula to find the number of non-negative integer solutions to this equation is given by:
\[ \binom{n + k - 1}{k - 1} \]
Substituting the values \( n = 4 \) and \( k = 3 \):
\[ \binom{4 + 3 - 1}{3 - 1} = \binom{6}{2} \]
Next, we can calculate \( \binom{6}{2} \):
\[ \binom{6}{2} = \frac{6!}{2!(6 - 2)!} = \frac{6 \times 5}{2 \times 1} = \frac{30}{2} = 15 \]
Thus, the total number of ways to place 4 indistinguishable balls into 3 distinguishable boxes is \( \boxed{15} \).