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Please forgive my lack of skill and terminology. I am attempting to find the number of possible combinations available for a video editing project.

There are four available video sources. Let's call them Box 1, Box 2, Box 3, and Box 4. Each of the boxes can be placed in one of four on-screen positions. Let's call the positions Left, Right, top, and Bottom.

Each of the combinations must use all four of the boxes. Each of the four positions can include anywhere from zero boxes to four boxes. An individual box, Box 1 for example, can't be assigned to two positions concurrently.

How many combinations are there?

First example combination: Box 1 Left , Box 2 Right , Box 3 Right , Box 4 Right

Second example combination: Box 1 Top , Box 2 Bottom , Box 3 Right , Box 4 Right

Third example combinations: Box 1 Right , Box 2 Right , Box 3 Right , Box 4 Right ,

Mark McCain
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  • Pick where box1 goes. There are four options. Pick where box2 goes. There are four options. Similarly pick where box3 and box4 go, four options each. Multiply the number of options available at each step to get the total count. $4^4=256$ Rule of Product – JMoravitz Aug 30 '21 at 17:59
  • Thank you. Is there a methodical way to build a matrix of the available combinations that shows which boxes occupy which positions? I'm trying to determine how I can account for each, without missing any. – Mark McCain Aug 30 '21 at 18:17
  • It is very simply a four-dimensional array with one of the four values in each position in the array. For just two boxes it looks like $\begin{array}{c|cccc}&L&R&T&B\\hline L&LL&LR<&LB\R&RL&RR&RT&RB\T&TL&TR&TT&TB\B&BL&BR&BT&BB\end{array}$ where the row corresponds to the first box's position and the column corresponds to the second box's position. Extend into a cube for adding the third box's position and into a hypercube for the fourth. – JMoravitz Aug 30 '21 at 18:30
  • Equivalently... you can think of this in terms of base-4 numbers... the first digit corresponding to which of the four positions box1 is in... the second digit corresponding to which of the four positions box2 is in etc... The possibilities being $0000,0001,0002,0003,0010,0011,\dots,3333$ – JMoravitz Aug 30 '21 at 18:31

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