25. Let
$$
S=\sum_{r=0}^{n}\binom{3 n+r}{r}
$$

Evaluate $S \div(23 \times 38 \times 41 \times 43 \times 47)$ when $n=12$.
25. Ans: 1274

By using the fact that $\binom{n}{r-1}+\binom{10}{r}=\binom{n+1}{r}$. we have
$$
\begin{aligned}
S & =\binom{3 n}{0}+\binom{3 n+1}{1}+\cdots+\binom{3 n+n}{n}=\binom{3 n+1}{0}+\binom{3 n+1}{1}+\cdots+\binom{3 n+n}{n} \\
& =\cdots=\binom{3 n+n}{n-1}+\binom{3 n+n}{n}=\binom{4 n+1}{n} .
\end{aligned}
$$

Thus when $n=12$.
$$
\frac{S}{23 \times 38 \times 41 \times 43 \times 47}=1274 \text {. }
$$