# 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 = $^8C_2 = 28$
• Combinations of 1 vertex and either I or J = $^8C_1 \cdot ^2C_1 = 16$
• Combinations between I and J = $1$

• Total = $28 + 16 + 1 = 45$

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.

• 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 $45-4 = 41$. Thanks – SilenceOnTheWire Oct 21 '16 at 17:59