number of ways to arrange pictures onto a wall I am stuck with this problem about combinations and permutations:
Lets suppose that I have $8$ frames, like the ones that hold a picture, each frame has its particular characteristics such as shape; we have $4$ available shapes. I would like to put inside these frames a set of $10$  family images, one image per frame; of course, that as long as I have less number of frames there would be some of them that will be without frame, but does not matter. Each frame can be painted in $5$ possible colours. At the end each frame with the picture can be put in any of $15$ different positions in a wall. The question is how many possibilities do I have to arrange those frames with the pictures in a wall considering the combination of shape, colour and image of each frame?
Any help? Should I do this exercise by pieces?
Thank you
 A: Suppose that we have ordered sequences of placeholders of length $8$; like this one below $$$$
$$\  \overline{\ \ 1\ \ } \ \overline{\ \ 2\ \ } \ \overline{\ \ 3\ \ } \ \overline{\ \ 4\ \ } \ \overline{\ \ 5\ \ } \ \overline{\ \ 6\ \ } \ \overline{\ \ 7 \ \ } \ \overline{\ \ 8\ \ } \ 
.$$ 
Such a sequence of placeholders represents a numbered sequence of colored and shaped frames. As an example, such a numbered sequence of frames can be described as follows: the first frame is a red square, the second frame is a green circle, ... , the eighth frame is a blue triangle.
We have $4^85^8$ such different numbered sequences of frames.
Also, we have $10$ distinguishable objects (photos) to fill the place holders up with. Taken one particular sequence of place holders, there are $10\times9\times8\times7\times6\times5\times4\times3$ possibilities to arrange the photos. As a total, we have $$4^8\times5^8\times10\times9\times8\times7\times6\times5\times4\times3$$ possibilities to order the photos. 
Now, there are $15$ ordered places on the wall. If we take one ordered sequence of frames with photos described above then the first thing we have to do is to choose empty places in which there will not be a framed photo. There are $$\begin{pmatrix}15 \\ 15-8\end{pmatrix}=\begin{pmatrix}15 \\ 7\end{pmatrix}$$ possibilities for the empty places on the wall. Given the empty places the rest is defined by the numbering of the frames. As a result the total number of the possible galleries is $$\begin{pmatrix}15 \\ 7\end{pmatrix}\times4^8\times5^8\times10\times9\times8\times7\times6\times5\times4\times3=298896998400000000000.
$$
A: We can choose the people to be excluded in ${10\choose2}$ ways. Then we can choose and paint the frames of the chosen people in $(4\cdot 5)^8$ ways. Order the framed people so obtained alphabetically; then place one picture after the other on the wall in $15\cdot14\cdot\ldots\cdot 8$ ways. After multiplying out this comes to the total given by zoli.
