Number of possible sequences

How many ways can two sets of objects, n_r of one type and n_l of another, be placed in N available spaces? The assumption is that each of the two types are separately indistinguishable. I.e. all n_r items look like one another as do the n_l objects. In the case of tossed coins, all heads are considered indistinguishable from each other as are all tails.

Suppose in a kindergarten there are N cubby holes where the children can place an orange or an apple. There are n_r oranges and n_l apples for a total n_r + n_l = N  total objects. The first cubby can hold any one of the N pieces of fruit, the second any of the remaining N - 1 pieces, the third any of N - 2, and so on until the last piece has only one place for it. The total number of possible arrangements is then N*(N-1)*(N-2)*...*1= N! ways. Of these ways however, many lead to the same result because the apples are undistinguishable, as are the oranges. The n_r oranges can be arranged in n_r ! different (but undistinguishable) ways and similarly n_l ! ways for the oranges, so a total of n_r ! *n_l ! possible but identical combinations must be divided out. The total remaining is then