$7 \cdot 77$ Given five functions:
(1) $f(x)=0, x \in R$;
(2) $f(x)=x^{2},-3<x \leqslant 3$;
(3) $f(x)=\log _{2}|x|, \quad x \neq 0, \quad x \in R$;
(4) $f(x)=(1+x)^{2 n}-(1-x)^{2 n}, \quad x \in R, n \in N$;
(5) $f(x)=\sin (\sin x), x \in R$.

Among them, the number of odd functions is
(A) 1.
(B) 2.
(C) 3.
(D) 4.
(2nd "Hope Cup" National Mathematics Invitational Competition, 1991)
[Solution] For (1): From $f(-x)=-f(x)=0$, then (1) can be regarded as an odd function;
For (4)
$$
\begin{aligned}
f(-x) & =[1+(-x)]^{2 n}-[1-(-x)]^{2 n} \\
& =-\left[(1+x)^{2 n}-(1-x)^{2 n}\right] \\
& =-f(x), \quad(x \in R, \quad n \in N)
\end{aligned}
$$

then (4) is an odd function.
And for (5)
$$
\begin{aligned}
f(-x) & =\sin [\sin (-x)]=\sin (-\sin x) \\
& =-\sin \sin x=-f(x)
\end{aligned}
$$

then (5) is an odd function.
Meanwhile, (2), (3) are not odd functions.
Therefore, the answer is $(C)$.