Have a following problem for which I'll show my reasoning (the problem is $1.6$ from book Problem solving methods in combinatorics by Pablo Soberon): If we want to write all the lists of length $n$ with integer elements $a_1,a_2,\ldots,a_n$ such that $|a_1|+ |a_2|+ \cdots + |a_n| \leq k$, and then to write all the lists with length $k$ with elements $b_1,b_2,\ldots,b_k$ such that $|b_1|+|b_2|+ \cdots + |b_k| \leq n$, prove that both lists have the same number of elements.
For the first one I thought I need to solve inequality $|a_1|+ |a_2|+ \cdots + |a_n| \leq k$ and to me it looked like if all $a_i$ $\geq 1$ then the number of solution should be $n+k-1 \choose n-1 $, since we need to choose $n$ elements from $k+1$ possibilities (using stars and bars method), and for other inequality that should be $n+k-1 \choose k-1$ which then doesn't look alright. Do you perhaps have any suggestion what am I missing in understanding?
Thank you for reading.
