How many lines can be created using the 6 vertices from a rectangular prism and 2 other points?

I got this prism

How many lines can be formed by the vertices of the prism and the
points I and J?

I did:

Combinations between the vertices = 8C2=28^8C_2 = 28
Combinations of 1 vertex and either I or J = 8C1⋅2C1=16^8C_1 \cdot ^2C_1 = 16
Combinations between I and J = 11
Total = 28+16+1=4528 + 16 + 1 = 45

But my book says the solution is 41. What did I do wrong?

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I admit the problem isn’t very clear. I am not sure if what is wanted is only the combinations between one vertex and either I or J or the combinations between all of them. I copied the problem as it is.

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1 Answer
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You double-counted a few lines. For instance, the line between C and J was already created by C and F, etc.

  

 

Ah, so to do this right I have to subtract all cases in which that happens, which are (C,J) , (C,I) , (D,I) and (F,J), so 45−4=4145-4 = 41. Thanks
– SilenceOnTheWire
2 days ago