We want to throw $n$ balls into $n$ bins uniformly and independently. Given these $n$ bins an order, what's the probability such that first $(k-1)$ bins contain $2$ balls and $k^{th}$ at least $(t-2)$ balls?
My attempt:
Let $A$ denotes the event such that first $(k-1)$ bins contains $2$ balls.
Let $B$ denote the event such that $n^{th}$ bin contains at least $(t-2)$ balls.
Then $P(A)={n \choose 2}(1-\frac{k-1}{n})^{n-2}(\frac{k-1}{n})^2$ and $P(B|A)$ is less than ${n-2 \choose t-2}(\frac{1}{n})^{t-2}$ by union bound.
$$P(A \cap B ) = P(A)P(B|A)\leq {n \choose 2}(1-\frac{k-1}{n})^{n-2}(\frac{k-1}{n})^2{n-2 \choose t-2}(\frac{1}{n})^{t-2}$$
but its looks quite weird, am I correct?