The answer is 120.

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A corner is a part of two plains of the original cube plus a triangular small plain. Two big original plain has eight corners so there will be 2 * 7 =14 surface diagonals out of which one where they meet is common to both. So

there are 13 suface diagonals. The diagonals of the triangular small plain are included in the big plains. Total diagonals from each corner are 24-1=23. So non surface diagonals from a corner are 23-13=10. There are 24 corners. So we counted 10 * 24=240. But a diagonal was counted from both the corners in two opposite directions. Diagonal is directionless. So 240/2=

120diagonals are through the solid.Or. Take the corner Gb. The sketch shows all 10 diagonals from Gb.