How do I find the number of solutions in the equation $$a_1 + a_2 + a_3 +\dots+a_k=2021$$ in terms of $k$ with $1\leq a_1, a_2, \dots, a_k\leq9$? I know the first step is to introduce a new variable, say $b_i + 1 = a_i$ for $i=1,2,\dots,k$ so that it turns into $$b_1+b_2+b_3+\dots b_k=2021-k$$ with $0\leq b_1, b_2,\dots, b_k\leq8$, but how do I continue?
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$\begingroup$ To complete the problem, just follow the model in this answer. I am voting to close, as a duplicate. $\endgroup$– user2661923Commented Mar 31, 2023 at 1:59
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$\begingroup$ With respect to the linked answer, in particular, notice the Addendum (shortcuts), which discuss capitalizing on the fact that each variable has the same upper bound. $\endgroup$– user2661923Commented Mar 31, 2023 at 2:02
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$\begingroup$ To the best of my knowledge, the only other approach is generating functions, which I am ignorant of. $\endgroup$– user2661923Commented Mar 31, 2023 at 2:04
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