Suppose we have an inexhaustible amount of black beads, white beads, and $n-k$ other colors of beads (red, blue, green, etc., etc.) (By "inexhaustible" it is meant that "at least $k$ black beads", "$k$ white beads", and "at least one bead of each of the $n-k$ other colors".)
How many ways/orders can we pick $n$ beads such that exactly $k$ beads are black or white and the remaining $n-k$ beads have no repeated colors?
To my understanding we can find how many different orders by doing the following. For the exactly $k$ white or black beads we have ${n \choose k}$ different ways to pick them. and $2^k$ ways to order them so ${n \choose k} 2^k$. For the second part to have no repeated other colors we would put $(n-k)$ choices for the first non black or white bead $(n-k-1)$ choices for the second, etc. bringing us to the falling factorial in terms of $n$ and $k$. so our answer would be ${n \choose k} 2^k (n-k)_k$ different ways. Is this correct? Would this be using a product principle for counting?