In a department store at the mall, black and brown gloves are on sale. There are $N$ (identical) pairs of black gloves and $N$ (identical) pairs of brown gloves. If $N$ customers come in, one at a time and randomly choose and buy $2$ pairs each, find the probability of event $A$: each customer buys $2$ pairs of different colors (one black and one brown).
My attempt: $2N$ trials, $N$ choices out of $N$ black gloves, $N$ choices out of $N$ brown gloves, but I don't know what to do next. Any help is appreciated!