$15 \cdot 33$ operation $x * y$ is defined as: $x * y=(x+1)(y+1)-1$, then which of the following statements is false?
(A) $x * y=y * x$ for all real values of $x$ and $y$.
(B) $x *(y+z)=(x * y)+(x * z)$ for all real values of $x, y$ and $z$.
(C) $(x-1) *(x+1)=(x * x)-1$ for all real values of $x$.
(D) $x * 0=x$ for all real values of $x$.
(E) $x *(y * z)=(x * y) * z$ for all real values of $x, y$ and $z$.
(33rd American High School Mathematics Examination, 1982)
[Solution] Given $x *(y+z)=(x+1)(y+z+1)-1$,
and
$$
\begin{aligned}
& (x * y)+(x * z) \\
= & {[(x+1)(y+1)-1]+[(x+1)(z+1)-1] } \\
= & (x+1)(y+z+2)-2,
\end{aligned}
$$

thus $x *(y+z) \neq(x * y)+(x * z)$.
Therefore, the answer is $(B)$.