carry out of full adder
You can write \$(A + B)\$ as \$A\cdot B + A \oplus B\$. 

Therefore:

\$C \cdot(A+B) + A\cdot B = C \cdot (A\oplus B) + C \cdot A \cdot B + A\cdot B =\\ = C \cdot (A\oplus B) + (1+C) \cdot A \cdot B = C \cdot (A\oplus B) + A \cdot B \$

The latter formula is the one you intuitively get, when you calculate the carry: you have a carry out if at least two inputs are 1.

The books use the latter formula because you don't need to calculate again \$A \oplus B\$, because you can reuse the sum term \$S=A\oplus B\$ (i.e. you save one logic gate).