I'm afraid your solution manuals is correct.
This is the Venn diagram of your problem.
What you want to do is to say P© without that region of intersection with A alone,B alone + the region of intersection of three circles.Right?
but when you say [math]P(A \cap C)[/math] you refer to the region in which A intersects C alone + that one in which the three circles intersect with each other.
And when you say [math] P(B \cap C) [/math] you also refer to the region in which B intersects with C alone + that one in which the three circles intersect.
If you added both values to each other with negative before them you'd be saying:It's the probability of C without regions of intersecion with A alone + B alone + 2 times the region of intersection of three circles.
That's why you compensate that by adding [math]P(A \cap B \cap C)[/math] with positive before it at the end of the equation.
I hope this helped