$35 \cdot 40$ A plane divides space into two parts, two planes can divide space into at most four parts. If four planes can divide space into $n$ parts at most, then $n$ equals
(A) 14 .
(B) 15 .
(C) 16 .
(D) 13 .
(E) 10 .
(China Beijing High School Mathematics Competition, 1984)
[Solution] Only when no two planes are parallel and no three planes intersect along the same line, can the considered planes divide the space into the maximum number of parts. Imagine the four faces of a tetrahedron extended into infinite planes; these four planes can serve as a "model". It is not difficult to calculate that four planes can divide the space into a maximum of 15 parts.
Therefore, the answer is $(B)$.
Note: Generally, it can be derived that:
$k$ planes can divide the space into a maximum of $\frac{k^{3}+5 k+6}{6}$ parts.