Question 6
Calculate the sum
$$T=\sum_{\substack{n_1,\ldots,n_{1994}\ge0\\n_1+2n_2+\cdots+1994n_{1994}=1994}}\frac{1}{n_1!n_2!\cdots n_{1994}!\,(n_2+2n_3+\cdots+1993n_{1994})!}.$$
The tuple entries are nonnegative integers, with $0!=1$. The source display omits the summation sign but immediately specifies that the expression is summed over all such tuples; the sign has been restored here.
$$T=\sum_{\substack{n_1,\ldots,n_{1994}\ge0\\n_1+2n_2+\cdots+1994n_{1994}=1994}}\frac{1}{n_1!n_2!\cdots n_{1994}!\,(n_2+2n_3+\cdots+1993n_{1994})!}.$$
The tuple entries are nonnegative integers, with $0!=1$. The source display omits the summation sign but immediately specifies that the expression is summed over all such tuples; the sign has been restored here.