The number of arrangements of six identical balls in three identical bins is ________.

Distributing $n = 6$ identical balls into $k = 3$ identical bins corresponds to finding the number of integer partitions of $6$ into at most $3$ parts (where bins can be empty).

Let the number of balls in the bins be $x_1 \ge x_2 \ge x_3 \ge 0$ such that $x_1 + x_2 + x_3 = 6$.

Listing all such partitions:

1. $6 + 0 + 0 = 6$

2. $5 + 1 + 0 = 6$

3. $4 + 2 + 0 = 6$

4. $4 + 1 + 1 = 6$

5. $3 + 3 + 0 = 6$

6. $3 + 2 + 1 = 6$

7. $2 + 2 + 2 = 6$

There are exactly 7 distinct arrangements.