Full adder carry expression
From your question it is unclear what $x$, $y$ and $z$ are. However, the two formulas are equivalent. Let me use products to denote a logical "AND" and additions to denote a logical "OR" so that your first formula becomes $xy + xz  + yz$

If you gather $z$ in the last two terms you get:
$xy + (x+y)z$.

Notice how the second term $(x+y)z$ is true iff $z$ is true, and at least one of $x$ and $y$ is true. However, if both $x$ and $y$ are true, the whole formula is true regardless of the truth value of the second term (since the first term is $xy$).
You can then replace $(x+y)$ with $(x \oplus x)$ (where "$\oplus$" denotes the exclusive OR) to obtain:
$xy + (x \oplus y)z$ or, equivalently, $xy + (x'y + xy')z$.