Choosing from Buckets

There are $N$ buckets numbered $1$ through $N$. The buckets contain balls; each ball has a color between $1$ and $K$. Let's denote the number of balls with color $j$ that are initially in bucket $i$ by $a_{i, j}$. For each $i$ from $1$ to $N-1$ (in this order), someone draws a ball uniformly at random from bucket $i$ and puts it into bucket $i+1$, then continues to draw the next ball. After putti