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 Question from jeff, a parent: We are trying to solve a combination problem. If I have 7 things that can be in 3 states (say on/partially on/off), and all 7 have to a state. How many different combinations can there be? It doesn't seem like we should treat it as 21 independent things, but not sure how to break the problem apart.

Hi Jeff,

I see this as 7 switches side by side on a wall, but these are not ordinary off-on switches. They each have three positions, off (f), on (n) and a middle position (m).

Consider the first switch on the left. you can place it in any one of three positions f, m, n. Regardless of which position you choose for the first switch the second switch can also be placed in any one of three positions. Thus the possible positions for the first two switches are

ff
fm
fn
mf
mm
mn
nf
nm
nn

Hence for the first two switches there are $3 \times 3 = 9$ possible positions. Again, regardless of the positions of the first two switches there are three possibilities for the position of the third switch.

Can you see now how to proceed?

Penny

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